| L(s) = 1 | − 4-s − 4·7-s − 9-s − 6·11-s − 12·13-s + 16-s + 6·17-s + 25-s + 4·28-s + 20·29-s + 36-s + 16·37-s − 22·43-s + 6·44-s + 16·47-s − 2·49-s + 12·52-s + 4·53-s − 10·59-s + 4·63-s − 64-s − 6·68-s + 24·77-s + 81-s − 10·89-s + 48·91-s + 6·97-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s − 1/3·9-s − 1.80·11-s − 3.32·13-s + 1/4·16-s + 1.45·17-s + 1/5·25-s + 0.755·28-s + 3.71·29-s + 1/6·36-s + 2.63·37-s − 3.35·43-s + 0.904·44-s + 2.33·47-s − 2/7·49-s + 1.66·52-s + 0.549·53-s − 1.30·59-s + 0.503·63-s − 1/8·64-s − 0.727·68-s + 2.73·77-s + 1/9·81-s − 1.05·89-s + 5.03·91-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5290975224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5290975224\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 53 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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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
−11.95130445620279555105292930615, −11.73268277348506822939164576404, −10.63107798155710850893054188665, −10.19627358390454237421692106133, −10.05910649385006350043135638057, −9.684401537888709125134557084602, −9.392790559505751737374961718262, −8.346004770478435630781846379377, −8.169665890612722472275159490918, −7.57629641220915144797087520316, −7.14875406008449061588844820898, −6.57537333988757274226815513754, −5.94000229741923408150142034328, −5.31461000195284040216294291539, −4.69288992423745143846298419014, −4.67652532162166858474477855511, −3.24219900064117815983168382880, −2.72151939447649770155378965519, −2.63761876189542316565364396671, −0.50219779347313353680217528982,
0.50219779347313353680217528982, 2.63761876189542316565364396671, 2.72151939447649770155378965519, 3.24219900064117815983168382880, 4.67652532162166858474477855511, 4.69288992423745143846298419014, 5.31461000195284040216294291539, 5.94000229741923408150142034328, 6.57537333988757274226815513754, 7.14875406008449061588844820898, 7.57629641220915144797087520316, 8.169665890612722472275159490918, 8.346004770478435630781846379377, 9.392790559505751737374961718262, 9.684401537888709125134557084602, 10.05910649385006350043135638057, 10.19627358390454237421692106133, 10.63107798155710850893054188665, 11.73268277348506822939164576404, 11.95130445620279555105292930615
