| L(s) = 1 | + 10·5-s − 13·7-s + 15·11-s − 10·13-s − 90·17-s + 86·19-s + 105·23-s + 25·25-s + 105·29-s + 71·31-s − 130·35-s − 298·37-s + 390·41-s − 169·43-s + 345·47-s + 636·49-s − 1.50e3·53-s + 150·55-s + 900·59-s − 61-s − 100·65-s + 335·67-s − 3.30e3·71-s + 314·73-s − 195·77-s − 346·79-s + 600·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.701·7-s + 0.411·11-s − 0.213·13-s − 1.28·17-s + 1.03·19-s + 0.951·23-s + 1/5·25-s + 0.672·29-s + 0.411·31-s − 0.627·35-s − 1.32·37-s + 1.48·41-s − 0.599·43-s + 1.07·47-s + 1.85·49-s − 3.88·53-s + 0.367·55-s + 1.98·59-s − 0.00209·61-s − 0.190·65-s + 0.610·67-s − 5.51·71-s + 0.503·73-s − 0.288·77-s − 0.492·79-s + 0.793·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.186134562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186134562\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 13 T - 467 T^{2} - 650 T^{3} + 228880 T^{4} - 650 p^{3} T^{5} - 467 p^{6} T^{6} + 13 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 15 T - 17 p T^{2} + 33750 T^{3} - 1901292 T^{4} + 33750 p^{3} T^{5} - 17 p^{7} T^{6} - 15 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 10 T - 2843 T^{2} - 14510 T^{3} + 3614740 T^{4} - 14510 p^{3} T^{5} - 2843 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 45 T + 8026 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 43 T - 1410 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 105 T - 13759 T^{2} - 47250 T^{3} + 332069592 T^{4} - 47250 p^{3} T^{5} - 13759 p^{6} T^{6} - 105 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 105 T - 38203 T^{2} - 47250 T^{3} + 1559683938 T^{4} - 47250 p^{3} T^{5} - 38203 p^{6} T^{6} - 105 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 71 T - 55709 T^{2} - 82928 T^{3} + 2652882388 T^{4} - 82928 p^{3} T^{5} - 55709 p^{6} T^{6} - 71 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 149 T + 86100 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 390 T + 59258 T^{2} + 17550000 T^{3} - 6613351377 T^{4} + 17550000 p^{3} T^{5} + 59258 p^{6} T^{6} - 390 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 169 T - 133073 T^{2} + 442780 T^{3} + 17533387480 T^{4} + 442780 p^{3} T^{5} - 133073 p^{6} T^{6} + 169 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 345 T - 60721 T^{2} + 9625500 T^{3} + 9171876612 T^{4} + 9625500 p^{3} T^{5} - 60721 p^{6} T^{6} - 345 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 750 T + 429154 T^{2} + 750 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 900 T + 233642 T^{2} - 149040000 T^{3} + 123651020523 T^{4} - 149040000 p^{3} T^{5} + 233642 p^{6} T^{6} - 900 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 214019 T^{2} - 239942 T^{3} - 5716040942 T^{4} - 239942 p^{3} T^{5} - 214019 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5 p T - 151217 T^{2} + 1690420 p T^{3} - 54809822480 T^{4} + 1690420 p^{4} T^{5} - 151217 p^{6} T^{6} - 5 p^{10} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1650 T + 1313422 T^{2} + 1650 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 157 T + 763440 T^{2} - 157 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 346 T - 564191 T^{2} - 104551166 T^{3} + 165616672204 T^{4} - 104551166 p^{3} T^{5} - 564191 p^{6} T^{6} + 346 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 600 T + 48926 T^{2} + 499500000 T^{3} - 436016659893 T^{4} + 499500000 p^{3} T^{5} + 48926 p^{6} T^{6} - 600 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1500 T + 1640338 T^{2} + 1500 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 623 T - 1456667 T^{2} - 12117350 T^{3} + 2216065417870 T^{4} - 12117350 p^{3} T^{5} - 1456667 p^{6} T^{6} - 623 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.18361117915084946375618251893, −6.12198286368618446369531816024, −6.10701127677770113096501931584, −5.49422713617261500627009608872, −5.39795969256695419862375046213, −5.37093620961798102707620693991, −5.21046667430749596021529660540, −4.74507234171340355237146428444, −4.40008949334100294605391071448, −4.32014530037584904282880562542, −4.26742653957870021174290829372, −3.94750271347046415311155274115, −3.44494147124954501111580353477, −3.32964440546093778314854683102, −3.17939559430296957432440393775, −2.69649888166424439177699129277, −2.56974559487338468981060365887, −2.39088322661713151520323247834, −2.31859486670811285564162551127, −1.51101031167891633535949411802, −1.33140479215579443712311261949, −1.30035199621454876271469190795, −1.14014571374305311618709831808, −0.30302195745513971927533375468, −0.17935480763125213467603313420,
0.17935480763125213467603313420, 0.30302195745513971927533375468, 1.14014571374305311618709831808, 1.30035199621454876271469190795, 1.33140479215579443712311261949, 1.51101031167891633535949411802, 2.31859486670811285564162551127, 2.39088322661713151520323247834, 2.56974559487338468981060365887, 2.69649888166424439177699129277, 3.17939559430296957432440393775, 3.32964440546093778314854683102, 3.44494147124954501111580353477, 3.94750271347046415311155274115, 4.26742653957870021174290829372, 4.32014530037584904282880562542, 4.40008949334100294605391071448, 4.74507234171340355237146428444, 5.21046667430749596021529660540, 5.37093620961798102707620693991, 5.39795969256695419862375046213, 5.49422713617261500627009608872, 6.10701127677770113096501931584, 6.12198286368618446369531816024, 6.18361117915084946375618251893
