| L(s) = 1 | − 2·3-s + 2·5-s − 9-s − 2·11-s − 10·13-s − 4·15-s − 10·17-s + 4·19-s + 4·23-s + 3·25-s + 6·27-s + 2·29-s − 12·31-s + 4·33-s − 4·37-s + 20·39-s − 4·41-s + 12·43-s − 2·45-s − 2·47-s + 20·51-s − 16·53-s − 4·55-s − 8·57-s − 4·59-s − 16·61-s − 20·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1/3·9-s − 0.603·11-s − 2.77·13-s − 1.03·15-s − 2.42·17-s + 0.917·19-s + 0.834·23-s + 3/5·25-s + 1.15·27-s + 0.371·29-s − 2.15·31-s + 0.696·33-s − 0.657·37-s + 3.20·39-s − 0.624·41-s + 1.82·43-s − 0.298·45-s − 0.291·47-s + 2.80·51-s − 2.19·53-s − 0.539·55-s − 1.05·57-s − 0.520·59-s − 2.04·61-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.631238679195046345978330328254, −9.436586401103624970366121579536, −9.102691490295267057013913864365, −8.817887005457252545901925322918, −8.001682359547254971705465851892, −7.54095536878199713415402242985, −7.07028403257897782882513009419, −6.97627950513349276641135508582, −6.10348288756034816076782637821, −6.05815683316642060527335324338, −5.22370344588509756500589218610, −5.18558606489697503301758832989, −4.69839196630239099295080704262, −4.34560793882724563776534950263, −3.12718643944308874880015815755, −2.79899269739276673639862887077, −2.24009351742173665139724222091, −1.63176368754881663258788704565, 0, 0,
1.63176368754881663258788704565, 2.24009351742173665139724222091, 2.79899269739276673639862887077, 3.12718643944308874880015815755, 4.34560793882724563776534950263, 4.69839196630239099295080704262, 5.18558606489697503301758832989, 5.22370344588509756500589218610, 6.05815683316642060527335324338, 6.10348288756034816076782637821, 6.97627950513349276641135508582, 7.07028403257897782882513009419, 7.54095536878199713415402242985, 8.001682359547254971705465851892, 8.817887005457252545901925322918, 9.102691490295267057013913864365, 9.436586401103624970366121579536, 9.631238679195046345978330328254
