| L(s) = 1 | − 2·5-s + 7-s + 11-s − 13-s + 2·17-s + 2·19-s − 5·23-s + 3·25-s + 29-s + 5·31-s − 2·35-s + 12·37-s − 12·41-s + 2·43-s + 13·47-s − 5·49-s − 3·53-s − 2·55-s − 10·59-s + 14·61-s + 2·65-s + 10·67-s + 3·71-s + 18·73-s + 77-s + 6·79-s + 20·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s − 1.04·23-s + 3/5·25-s + 0.185·29-s + 0.898·31-s − 0.338·35-s + 1.97·37-s − 1.87·41-s + 0.304·43-s + 1.89·47-s − 5/7·49-s − 0.412·53-s − 0.269·55-s − 1.30·59-s + 1.79·61-s + 0.248·65-s + 1.22·67-s + 0.356·71-s + 2.10·73-s + 0.113·77-s + 0.675·79-s + 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.726965735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.726965735\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 50 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 128 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 134 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.823515612491441912882040047812, −8.282241270321047635030709090503, −7.944951697660732035604283625654, −7.904577922833787535071223303594, −7.53381858260770277969111023531, −6.95279808740597754452538918287, −6.55361472679277518570412389505, −6.38648965430998209541156520096, −5.77261846961797279458684752678, −5.39933642400471517089722842931, −4.89916663875080928341708038177, −4.66227671868133182889211176051, −4.07192072118069796420717186976, −3.86123229693247741638981815478, −3.24813357152975580898265374995, −2.99568365351038639136920216979, −2.06133784547470349776681212413, −2.05151696499587778666880325446, −0.836615048177461162379251488877, −0.71987161036480443378039294940,
0.71987161036480443378039294940, 0.836615048177461162379251488877, 2.05151696499587778666880325446, 2.06133784547470349776681212413, 2.99568365351038639136920216979, 3.24813357152975580898265374995, 3.86123229693247741638981815478, 4.07192072118069796420717186976, 4.66227671868133182889211176051, 4.89916663875080928341708038177, 5.39933642400471517089722842931, 5.77261846961797279458684752678, 6.38648965430998209541156520096, 6.55361472679277518570412389505, 6.95279808740597754452538918287, 7.53381858260770277969111023531, 7.904577922833787535071223303594, 7.944951697660732035604283625654, 8.282241270321047635030709090503, 8.823515612491441912882040047812
