| L(s) = 1 | + 4·3-s + 8·9-s − 2·16-s − 24·17-s − 12·19-s + 16·23-s + 12·27-s − 4·31-s − 4·37-s + 12·41-s + 20·43-s + 16·47-s − 8·48-s − 96·51-s − 48·57-s + 4·67-s + 64·69-s − 44·71-s − 28·73-s + 8·79-s + 9·81-s + 28·83-s − 40·89-s − 16·93-s + 8·97-s − 8·101-s + 24·103-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 1/2·16-s − 5.82·17-s − 2.75·19-s + 3.33·23-s + 2.30·27-s − 0.718·31-s − 0.657·37-s + 1.87·41-s + 3.04·43-s + 2.33·47-s − 1.15·48-s − 13.4·51-s − 6.35·57-s + 0.488·67-s + 7.70·69-s − 5.22·71-s − 3.27·73-s + 0.900·79-s + 81-s + 3.07·83-s − 4.23·89-s − 1.65·93-s + 0.812·97-s − 0.796·101-s + 2.36·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.223726464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.223726464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 16 T + 128 T^{2} - 816 T^{3} + 4418 T^{4} - 816 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + 23 T^{4} - 20 p T^{5} + 128 T^{6} - 224 T^{7} + 385 T^{8} - 224 p T^{9} + 128 p^{2} T^{10} - 20 p^{4} T^{11} + 23 p^{4} T^{12} - 4 p^{6} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 8 T^{3} + 47 T^{4} - 16 p T^{5} + 32 T^{6} - 400 T^{7} + 393 T^{8} - 400 p T^{9} + 32 p^{2} T^{10} - 16 p^{4} T^{11} + 47 p^{4} T^{12} + 8 p^{5} T^{13} + p^{8} T^{16} \) |
| 11 | \( 1 - 38 T^{2} + 559 T^{4} - 3624 T^{6} + 16613 T^{8} - 3624 p^{2} T^{10} + 559 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 60 T^{3} - 9 p T^{4} + 480 T^{5} + 1800 T^{6} + 360 T^{7} - 14095 T^{8} + 360 p T^{9} + 1800 p^{2} T^{10} + 480 p^{3} T^{11} - 9 p^{5} T^{12} - 60 p^{5} T^{13} + p^{8} T^{16} \) |
| 17 | \( 1 + 24 T + 288 T^{2} + 2492 T^{3} + 17875 T^{4} + 109496 T^{5} + 2024 p^{2} T^{6} + 165224 p T^{7} + 12206433 T^{8} + 165224 p^{2} T^{9} + 2024 p^{4} T^{10} + 109496 p^{3} T^{11} + 17875 p^{4} T^{12} + 2492 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 6 T + 79 T^{2} + 308 T^{3} + 2231 T^{4} + 308 p T^{5} + 79 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 168 T^{2} + 13172 T^{4} - 646200 T^{6} + 22104454 T^{8} - 646200 p^{2} T^{10} + 13172 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 2 T + 89 T^{2} + 100 T^{3} + 3547 T^{4} + 100 p T^{5} + 89 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 4 T + 8 T^{2} - 204 T^{3} - 1720 T^{4} + 1420 T^{5} + 40248 T^{6} + 120124 T^{7} + 1051038 T^{8} + 120124 p T^{9} + 40248 p^{2} T^{10} + 1420 p^{3} T^{11} - 1720 p^{4} T^{12} - 204 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 6 T + 159 T^{2} - 16 p T^{3} + 9557 T^{4} - 16 p^{2} T^{5} + 159 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 20 T + 200 T^{2} - 1020 T^{3} - 808 T^{4} + 33220 T^{5} + 17400 T^{6} - 3432980 T^{7} + 34938366 T^{8} - 3432980 p T^{9} + 17400 p^{2} T^{10} + 33220 p^{3} T^{11} - 808 p^{4} T^{12} - 1020 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 16 T + 128 T^{2} - 1280 T^{3} + 16012 T^{4} - 128416 T^{5} + 824320 T^{6} - 6998448 T^{7} + 58173798 T^{8} - 6998448 p T^{9} + 824320 p^{2} T^{10} - 128416 p^{3} T^{11} + 16012 p^{4} T^{12} - 1280 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 800 T^{3} + 1572 T^{4} + 40800 T^{5} + 320000 T^{6} - 712000 T^{7} - 23620442 T^{8} - 712000 p T^{9} + 320000 p^{2} T^{10} + 40800 p^{3} T^{11} + 1572 p^{4} T^{12} - 800 p^{5} T^{13} + p^{8} T^{16} \) |
| 59 | \( 1 - 364 T^{2} + 63104 T^{4} - 6721844 T^{6} + 479355310 T^{8} - 6721844 p^{2} T^{10} + 63104 p^{4} T^{12} - 364 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 150 T^{2} + 15659 T^{4} - 1211524 T^{6} + 84989469 T^{8} - 1211524 p^{2} T^{10} + 15659 p^{4} T^{12} - 150 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 4 T + 8 T^{2} - 1628 T^{3} + 8792 T^{4} + 15364 T^{5} + 1193400 T^{6} - 8217060 T^{7} - 18996674 T^{8} - 8217060 p T^{9} + 1193400 p^{2} T^{10} + 15364 p^{3} T^{11} + 8792 p^{4} T^{12} - 1628 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 22 T + 351 T^{2} + 4092 T^{3} + 38567 T^{4} + 4092 p T^{5} + 351 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 28 T + 392 T^{2} + 3628 T^{3} + 22344 T^{4} + 100612 T^{5} + 120 p^{2} T^{6} + 95508 p T^{7} + 68649790 T^{8} + 95508 p^{2} T^{9} + 120 p^{4} T^{10} + 100612 p^{3} T^{11} + 22344 p^{4} T^{12} + 3628 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 4 T + 106 T^{2} + 1132 T^{3} - 1082 T^{4} + 1132 p T^{5} + 106 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 28 T + 392 T^{2} - 3396 T^{3} + 14600 T^{4} + 4508 T^{5} - 83016 T^{6} - 6317244 T^{7} + 96756958 T^{8} - 6317244 p T^{9} - 83016 p^{2} T^{10} + 4508 p^{3} T^{11} + 14600 p^{4} T^{12} - 3396 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 20 T + 306 T^{2} + 3340 T^{3} + 36126 T^{4} + 3340 p T^{5} + 306 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 8 T + 32 T^{2} - 156 T^{3} - 19525 T^{4} + 113288 T^{5} - 269336 T^{6} - 3591384 T^{7} + 219989137 T^{8} - 3591384 p T^{9} - 269336 p^{2} T^{10} + 113288 p^{3} T^{11} - 19525 p^{4} T^{12} - 156 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.20784541634049102588687576820, −4.20450165161323239139220770516, −4.10239985161203829525576983699, −3.97365987826924814996488571156, −3.71660534450092498697088137615, −3.28565586889136592877047776386, −3.24227493931035848136814603049, −3.21072526924433451610156137526, −3.11134320811390335476426906765, −3.09105975406390065932590383241, −2.69564758856068329176576973771, −2.55194742821226190455392546505, −2.52749092981393555878103933992, −2.49107800922481888036498165229, −2.41014060770720824634205567224, −1.99740560016824840860067901975, −1.98523155975036123229409418223, −1.95008719699737186397160142424, −1.91364239253395040573470092301, −1.48374683932477967711703401201, −1.33867766522752495858730049204, −1.03767507350411313691777502112, −0.65510549360242059568550023036, −0.40485668785958006269189348062, −0.37295170269447312768187775755,
0.37295170269447312768187775755, 0.40485668785958006269189348062, 0.65510549360242059568550023036, 1.03767507350411313691777502112, 1.33867766522752495858730049204, 1.48374683932477967711703401201, 1.91364239253395040573470092301, 1.95008719699737186397160142424, 1.98523155975036123229409418223, 1.99740560016824840860067901975, 2.41014060770720824634205567224, 2.49107800922481888036498165229, 2.52749092981393555878103933992, 2.55194742821226190455392546505, 2.69564758856068329176576973771, 3.09105975406390065932590383241, 3.11134320811390335476426906765, 3.21072526924433451610156137526, 3.24227493931035848136814603049, 3.28565586889136592877047776386, 3.71660534450092498697088137615, 3.97365987826924814996488571156, 4.10239985161203829525576983699, 4.20450165161323239139220770516, 4.20784541634049102588687576820
Plot not available for L-functions of degree greater than 10.
