| L(s) = 1 | − 9·4-s + 17·16-s + 96·17-s − 288·23-s − 240·25-s + 384·29-s + 1.28e3·43-s − 592·49-s − 984·53-s + 288·61-s − 153·64-s − 864·68-s + 4.32e3·79-s + 2.59e3·92-s + 2.16e3·100-s + 1.19e3·103-s + 2.49e3·107-s − 3.45e3·116-s − 2.17e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 9/8·4-s + 0.265·16-s + 1.36·17-s − 2.61·23-s − 1.91·25-s + 2.45·29-s + 4.56·43-s − 1.72·49-s − 2.55·53-s + 0.604·61-s − 0.298·64-s − 1.54·68-s + 6.15·79-s + 2.93·92-s + 2.15·100-s + 1.14·103-s + 2.25·107-s − 2.76·116-s − 1.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.611995071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.611995071\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 9 T^{2} + p^{6} T^{4} + 9 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 48 p T^{2} + 44302 T^{4} + 48 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 592 T^{2} + 310782 T^{4} + 592 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 2172 T^{2} + 3811270 T^{4} + 2172 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 48 T - 1730 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 13552 T^{2} + 94862766 T^{4} + 13552 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 192 T + 45862 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 18592 T^{2} + 1722523710 T^{4} + 18592 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 92356 T^{2} + 6285341910 T^{4} + 92356 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 142320 T^{2} + 14548520830 T^{4} + 142320 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 644 T + 250566 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 385164 T^{2} + 58535996374 T^{4} + 385164 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 492 T + 164158 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 758940 T^{2} + 227837028550 T^{4} + 758940 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 144 T + 393094 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 611104 T^{2} + 187596510414 T^{4} + 611104 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 514188 T^{2} + 171350572726 T^{4} + 514188 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 659668 T^{2} + 389934034566 T^{4} + 659668 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2160 T + 2130910 T^{2} - 2160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 823068 T^{2} + 604381635622 T^{4} + 823068 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 2258064 T^{2} + 2268670823038 T^{4} + 2258064 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 3513652 T^{2} + 4748412207846 T^{4} + 3513652 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.27761059688710239072835316572, −6.03227056226892833532949985796, −5.99338064002945197627478154690, −5.82243433727049161973506361052, −5.59157550818188078345752440370, −5.05595570024576076584647712644, −4.95161516443529261689380002837, −4.92789887017885145648806710783, −4.75565083510645587656220721721, −4.11207453878641751891100709910, −4.07781631222911387694578150741, −3.97478438547376408660822377134, −3.91381682397942018357553417481, −3.47099208295994480516293997481, −3.18473139534580197253669217107, −2.81321100793184268722843225666, −2.71586862987272771262392252407, −2.37907505748095102302664250422, −1.92855312603084206193136482107, −1.88749426711453453609746327506, −1.48814742020961620584734487152, −1.09659231418290510296333234175, −0.66467533455827391312761878823, −0.54155455502002189805770486356, −0.30225270113315846921080493552,
0.30225270113315846921080493552, 0.54155455502002189805770486356, 0.66467533455827391312761878823, 1.09659231418290510296333234175, 1.48814742020961620584734487152, 1.88749426711453453609746327506, 1.92855312603084206193136482107, 2.37907505748095102302664250422, 2.71586862987272771262392252407, 2.81321100793184268722843225666, 3.18473139534580197253669217107, 3.47099208295994480516293997481, 3.91381682397942018357553417481, 3.97478438547376408660822377134, 4.07781631222911387694578150741, 4.11207453878641751891100709910, 4.75565083510645587656220721721, 4.92789887017885145648806710783, 4.95161516443529261689380002837, 5.05595570024576076584647712644, 5.59157550818188078345752440370, 5.82243433727049161973506361052, 5.99338064002945197627478154690, 6.03227056226892833532949985796, 6.27761059688710239072835316572
