| L(s) = 1 | − 8·4-s − 4·9-s − 8·13-s + 28·16-s + 24·19-s + 20·25-s + 32·36-s − 8·43-s + 16·47-s + 32·49-s + 64·52-s − 16·53-s − 48·59-s − 48·64-s + 16·67-s − 192·76-s + 10·81-s − 16·83-s + 32·89-s − 160·100-s − 56·103-s + 32·117-s + 60·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4·4-s − 4/3·9-s − 2.21·13-s + 7·16-s + 5.50·19-s + 4·25-s + 16/3·36-s − 1.21·43-s + 2.33·47-s + 32/7·49-s + 8.87·52-s − 2.19·53-s − 6.24·59-s − 6·64-s + 1.95·67-s − 22.0·76-s + 10/9·81-s − 1.75·83-s + 3.39·89-s − 16·100-s − 5.51·103-s + 2.95·117-s + 5.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3713064666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3713064666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{4} \) |
| 17 | \( 1 \) |
| good | 2 | \( ( 1 + p^{2} T^{2} + 5 p T^{4} + p^{4} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 4 p T^{2} + 202 T^{4} - 1392 T^{6} + 7571 T^{8} - 1392 p^{2} T^{10} + 202 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 8 T + 16 T^{2} + 8 p T^{3} - 320 T^{4} + 8 p^{2} T^{5} + 16 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 16 T^{2} - 8 p T^{3} - 320 T^{4} - 8 p^{2} T^{5} + 16 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 - 60 T^{2} + 1610 T^{4} - 26768 T^{6} + 329715 T^{8} - 26768 p^{2} T^{10} + 1610 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 4 T + 38 T^{2} + 88 T^{3} + 603 T^{4} + 88 p T^{5} + 38 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 12 T + 94 T^{2} - 512 T^{3} + 2483 T^{4} - 512 p T^{5} + 94 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 68 T^{2} + 1626 T^{4} + 4496 T^{6} - 758653 T^{8} + 4496 p^{2} T^{10} + 1626 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 72 T^{2} + 2690 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 88 T^{2} + 3788 T^{4} - 87720 T^{6} + 1953702 T^{8} - 87720 p^{2} T^{10} + 3788 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 112 T^{2} + 6592 T^{4} - 273264 T^{6} + 10269026 T^{8} - 273264 p^{2} T^{10} + 6592 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 180 T^{2} + 10762 T^{4} - 144464 T^{6} - 5175757 T^{8} - 144464 p^{2} T^{10} + 10762 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T + 94 T^{2} + 320 T^{3} + 5683 T^{4} + 320 p T^{5} + 94 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 8 T + 116 T^{2} - 520 T^{3} + 122 p T^{4} - 520 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8 T + 156 T^{2} + 1112 T^{3} + 11414 T^{4} + 1112 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 24 T + 432 T^{2} + 4856 T^{3} + 44528 T^{4} + 4856 p T^{5} + 432 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 176 T^{2} + 10688 T^{4} - 54704 T^{6} - 19549662 T^{8} - 54704 p^{2} T^{10} + 10688 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 8 T + 272 T^{2} - 1592 T^{3} + 27474 T^{4} - 1592 p T^{5} + 272 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 320 T^{2} + 46404 T^{4} - 4296640 T^{6} + 320522054 T^{8} - 4296640 p^{2} T^{10} + 46404 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 24 T^{2} - 1828 T^{4} - 356136 T^{6} + 33920518 T^{8} - 356136 p^{2} T^{10} - 1828 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 528 T^{2} + 128032 T^{4} - 18675216 T^{6} + 1795341570 T^{8} - 18675216 p^{2} T^{10} + 128032 p^{4} T^{12} - 528 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 8 T + 212 T^{2} + 1576 T^{3} + 21142 T^{4} + 1576 p T^{5} + 212 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + 336 T^{2} - 3376 T^{3} + 41408 T^{4} - 3376 p T^{5} + 336 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 600 T^{2} + 168524 T^{4} - 29080744 T^{6} + 3388928166 T^{8} - 29080744 p^{2} T^{10} + 168524 p^{4} T^{12} - 600 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.46435876245679325979525878965, −4.41331181142314171296016522158, −4.20005996886951056415800298007, −4.14344399264552109897167929659, −3.96017989237060562447032937124, −3.63986020671770263565645025454, −3.46303924148618432821873452135, −3.32602579314129536721391761449, −3.26092369444195376862433782365, −3.22614756992582128076243598704, −3.08010863433071741727888849057, −3.04533838513688000142540107328, −2.88772139169852969958695876457, −2.59971387892743302803092863763, −2.42739875605536125469984088328, −2.32255779525893652404522640468, −2.04303396576816927206311968660, −1.96616624295140716116109857962, −1.43432411820251835627738237818, −1.27000144443455256777091019618, −1.11578897178307063611908771445, −0.837117857915202922063444844805, −0.77022897411635701593274219856, −0.56278967619491750515209592328, −0.14766586940841130149998939347,
0.14766586940841130149998939347, 0.56278967619491750515209592328, 0.77022897411635701593274219856, 0.837117857915202922063444844805, 1.11578897178307063611908771445, 1.27000144443455256777091019618, 1.43432411820251835627738237818, 1.96616624295140716116109857962, 2.04303396576816927206311968660, 2.32255779525893652404522640468, 2.42739875605536125469984088328, 2.59971387892743302803092863763, 2.88772139169852969958695876457, 3.04533838513688000142540107328, 3.08010863433071741727888849057, 3.22614756992582128076243598704, 3.26092369444195376862433782365, 3.32602579314129536721391761449, 3.46303924148618432821873452135, 3.63986020671770263565645025454, 3.96017989237060562447032937124, 4.14344399264552109897167929659, 4.20005996886951056415800298007, 4.41331181142314171296016522158, 4.46435876245679325979525878965
Plot not available for L-functions of degree greater than 10.
