| L(s) = 1 | + 9·2-s + 19·4-s − 108·7-s − 99·8-s − 168·11-s − 1.29e3·13-s − 972·14-s − 501·16-s + 576·17-s − 1.33e3·19-s − 1.51e3·22-s + 5.90e3·23-s − 1.16e4·26-s − 2.05e3·28-s + 3.55e3·29-s − 1.16e4·31-s − 2.11e3·32-s + 5.18e3·34-s − 1.46e4·37-s − 1.20e4·38-s − 1.81e3·41-s + 7.56e3·43-s − 3.19e3·44-s + 5.31e4·46-s + 3.24e3·47-s − 1.20e4·49-s − 2.46e4·52-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.593·4-s − 0.833·7-s − 0.546·8-s − 0.418·11-s − 2.12·13-s − 1.32·14-s − 0.489·16-s + 0.483·17-s − 0.849·19-s − 0.666·22-s + 2.32·23-s − 3.38·26-s − 0.494·28-s + 0.784·29-s − 2.17·31-s − 0.365·32-s + 0.769·34-s − 1.76·37-s − 1.35·38-s − 0.168·41-s + 0.623·43-s − 0.248·44-s + 3.70·46-s + 0.213·47-s − 0.716·49-s − 1.26·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 9 T + 31 p T^{2} - 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 108 T + 23714 T^{2} + 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 168 T + 69634 T^{2} + 168 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1296 T + 1005494 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 576 T + 2486558 T^{2} - 576 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1336 T + 5283078 T^{2} + 1336 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5904 T + 21414686 T^{2} - 5904 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3552 T + 27566938 T^{2} - 3552 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11648 T + 90715902 T^{2} + 11648 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14688 T + 189133094 T^{2} + 14688 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1812 T + 29066422 T^{2} + 1812 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7560 T + 268114310 T^{2} - 7560 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3240 T + 424402910 T^{2} - 3240 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 22176 T + 869263526 T^{2} + 22176 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 57336 T + 2002300258 T^{2} + 57336 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 30140 T + 1692991518 T^{2} + 30140 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5184 T + 2244311078 T^{2} - 5184 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17424 T + 3450786046 T^{2} + 17424 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3456 T + 4135172546 T^{2} - 3456 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 57520 T + 5031936798 T^{2} + 57520 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 133992 T + 11327999318 T^{2} - 133992 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 136764 T + 15341778358 T^{2} + 136764 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 85536 T + 13589022338 T^{2} + 85536 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07050330239817598146237762777, −10.82844643476686734777346664312, −10.24578492055392414691644532616, −9.673103082384324424610991561043, −9.104879892942790354407768263439, −8.970424239318706673926161403933, −7.948799417236046809789091409708, −7.40937732158371478272968321438, −6.95970156309237975246417622501, −6.48549105332996040294271121706, −5.52051416122668004953257340565, −5.38294515058000126829766285449, −4.61117391144864773976846805699, −4.54192644970125128986069695296, −3.32995927564655044851254480784, −3.24177029129062353982154712161, −2.47547932227067708239122647207, −1.50633019787292021121762362658, 0, 0,
1.50633019787292021121762362658, 2.47547932227067708239122647207, 3.24177029129062353982154712161, 3.32995927564655044851254480784, 4.54192644970125128986069695296, 4.61117391144864773976846805699, 5.38294515058000126829766285449, 5.52051416122668004953257340565, 6.48549105332996040294271121706, 6.95970156309237975246417622501, 7.40937732158371478272968321438, 7.948799417236046809789091409708, 8.970424239318706673926161403933, 9.104879892942790354407768263439, 9.673103082384324424610991561043, 10.24578492055392414691644532616, 10.82844643476686734777346664312, 11.07050330239817598146237762777
