| L(s) = 1 | + 54·5-s + 88·7-s + 540·11-s + 418·13-s − 1.18e3·17-s + 1.67e3·19-s − 4.10e3·23-s + 3.12e3·25-s − 594·29-s − 4.25e3·31-s + 4.75e3·35-s − 596·37-s + 1.72e4·41-s + 1.21e4·43-s − 1.29e3·47-s + 1.68e4·49-s − 3.89e4·53-s + 2.91e4·55-s − 7.66e3·59-s + 3.47e4·61-s + 2.25e4·65-s − 2.18e4·67-s + 9.37e4·71-s + 1.35e5·73-s + 4.75e4·77-s + 7.69e4·79-s + 6.77e4·83-s + ⋯ |
| L(s) = 1 | + 0.965·5-s + 0.678·7-s + 1.34·11-s + 0.685·13-s − 0.996·17-s + 1.06·19-s − 1.61·23-s + 25-s − 0.131·29-s − 0.795·31-s + 0.655·35-s − 0.0715·37-s + 1.60·41-s + 0.997·43-s − 0.0855·47-s + 49-s − 1.90·53-s + 1.29·55-s − 0.286·59-s + 1.19·61-s + 0.662·65-s − 0.593·67-s + 2.20·71-s + 2.96·73-s + 0.913·77-s + 1.38·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.604977793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.604977793\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 54 T - 209 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 88 T - 9063 T^{2} - 88 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 540 T + 130549 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 418 T - 196569 T^{2} - 418 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 594 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 44 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4104 T + 10406473 T^{2} + 4104 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 594 T - 20158313 T^{2} + 594 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 4256 T - 10515615 T^{2} + 4256 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 298 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 17226 T + 180878875 T^{2} - 17226 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12100 T - 598443 T^{2} - 12100 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1296 T - 227665391 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 19494 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7668 T - 656126075 T^{2} + 7668 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34738 T + 362132343 T^{2} - 34738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 21812 T - 874361763 T^{2} + 21812 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 67562 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 76912 T + 2838399345 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 67716 T + 646416013 T^{2} - 67716 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 29754 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122398 T + 6393930147 T^{2} - 122398 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.92649702607682311703254955277, −10.89887011771914365320340156175, −9.807943831200822130288449857749, −9.655934411275165945831790647707, −9.187074444656611724322628695851, −8.828681445213981244861034284480, −8.139075808496025515233764145686, −7.80154171127378889501329440786, −7.07360021684453199247855318453, −6.58951439953734889794889125230, −6.07879253362114707887642589915, −5.76077477337314444184524114433, −5.04225698676996077440506787299, −4.50405718249056239814185247069, −3.82542783912059056199759849929, −3.42644155167109205238045327849, −2.14928993905842677493514034177, −2.12434343778050494847770796435, −1.17859324885779690518017578175, −0.68763713007213654333199162349,
0.68763713007213654333199162349, 1.17859324885779690518017578175, 2.12434343778050494847770796435, 2.14928993905842677493514034177, 3.42644155167109205238045327849, 3.82542783912059056199759849929, 4.50405718249056239814185247069, 5.04225698676996077440506787299, 5.76077477337314444184524114433, 6.07879253362114707887642589915, 6.58951439953734889794889125230, 7.07360021684453199247855318453, 7.80154171127378889501329440786, 8.139075808496025515233764145686, 8.828681445213981244861034284480, 9.187074444656611724322628695851, 9.655934411275165945831790647707, 9.807943831200822130288449857749, 10.89887011771914365320340156175, 10.92649702607682311703254955277
