| L(s) = 1 | + 1.67·2-s + 0.806·4-s − 3.28·5-s − 2.15·7-s − 1.99·8-s − 5.50·10-s − 4.63·11-s + 2.19·13-s − 3.61·14-s − 4.96·16-s − 6.31·17-s − 1.80·19-s − 2.64·20-s − 7.76·22-s − 7.35·23-s + 5.80·25-s + 3.67·26-s − 1.73·28-s + 0.0630·29-s − 4.31·32-s − 10.5·34-s + 7.08·35-s − 3.93·37-s − 3.02·38-s + 6.57·40-s − 4.24·41-s − 4.38·43-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.403·4-s − 1.47·5-s − 0.815·7-s − 0.707·8-s − 1.74·10-s − 1.39·11-s + 0.608·13-s − 0.965·14-s − 1.24·16-s − 1.53·17-s − 0.414·19-s − 0.592·20-s − 1.65·22-s − 1.53·23-s + 1.16·25-s + 0.720·26-s − 0.328·28-s + 0.0116·29-s − 0.762·32-s − 1.81·34-s + 1.19·35-s − 0.646·37-s − 0.490·38-s + 1.03·40-s − 0.663·41-s − 0.668·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1793842401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1793842401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 7.35T + 23T^{2} \) |
| 29 | \( 1 - 0.0630T + 29T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 4.38T + 43T^{2} \) |
| 47 | \( 1 + 6.37T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 8.96T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 + 9.73T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 7.28T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 + 5.38T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73006776213424567202920403908, −6.82946320755333069293134690110, −6.41665076473770090537216716515, −5.53157013977944831550197569219, −4.87037649953355045229342204521, −4.07612703149625545238680464713, −3.75577003511591961690901535288, −2.95942162942757360089865250796, −2.18698859591796869776207653653, −0.15734974231965045334247232864,
0.15734974231965045334247232864, 2.18698859591796869776207653653, 2.95942162942757360089865250796, 3.75577003511591961690901535288, 4.07612703149625545238680464713, 4.87037649953355045229342204521, 5.53157013977944831550197569219, 6.41665076473770090537216716515, 6.82946320755333069293134690110, 7.73006776213424567202920403908
