| L(s) = 1 | + 2·5-s − 4·9-s + 4·11-s − 4·13-s + 4·17-s − 2·19-s − 8·23-s + 3·25-s − 4·29-s − 12·37-s + 4·41-s − 8·45-s − 6·49-s − 12·53-s + 8·55-s − 8·61-s − 8·65-s − 16·67-s − 8·71-s − 4·73-s + 24·79-s + 7·81-s + 16·83-s + 8·85-s − 12·89-s − 4·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 4/3·9-s + 1.20·11-s − 1.10·13-s + 0.970·17-s − 0.458·19-s − 1.66·23-s + 3/5·25-s − 0.742·29-s − 1.97·37-s + 0.624·41-s − 1.19·45-s − 6/7·49-s − 1.64·53-s + 1.07·55-s − 1.02·61-s − 0.992·65-s − 1.95·67-s − 0.949·71-s − 0.468·73-s + 2.70·79-s + 7/9·81-s + 1.75·83-s + 0.867·85-s − 1.27·89-s − 0.410·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 196 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 180 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.78513685174352828177882674767, −7.76760565092107050697203067396, −6.97069244630425161000059216205, −6.96377445862106700441733988411, −6.23975023577786343083606399407, −6.15017307823796868359685993137, −5.87689702938886043946719670477, −5.41413550682614316386569133086, −5.11120029313900633291150555543, −4.69432521342562656719993507575, −4.28059413026400237982553779563, −3.75934419873765209491993547512, −3.30192175563577018740494259590, −3.13566745663642548051411453896, −2.37460344817134277853307979169, −2.20660523819540497949679362949, −1.54349588144185699979579775229, −1.29501968330739158341611946759, 0, 0,
1.29501968330739158341611946759, 1.54349588144185699979579775229, 2.20660523819540497949679362949, 2.37460344817134277853307979169, 3.13566745663642548051411453896, 3.30192175563577018740494259590, 3.75934419873765209491993547512, 4.28059413026400237982553779563, 4.69432521342562656719993507575, 5.11120029313900633291150555543, 5.41413550682614316386569133086, 5.87689702938886043946719670477, 6.15017307823796868359685993137, 6.23975023577786343083606399407, 6.96377445862106700441733988411, 6.97069244630425161000059216205, 7.76760565092107050697203067396, 7.78513685174352828177882674767
