| L(s) = 1 | − 1.84·2-s + 2.98·3-s + 1.39·4-s − 5.49·6-s + 5.05·7-s + 1.11·8-s + 5.90·9-s − 3.23·11-s + 4.15·12-s − 2.63·13-s − 9.31·14-s − 4.84·16-s + 0.962·17-s − 10.8·18-s − 2.66·19-s + 15.0·21-s + 5.95·22-s + 5.80·23-s + 3.34·24-s + 4.85·26-s + 8.68·27-s + 7.03·28-s + 2.13·29-s − 6.68·31-s + 6.68·32-s − 9.64·33-s − 1.77·34-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 1.72·3-s + 0.696·4-s − 2.24·6-s + 1.91·7-s + 0.395·8-s + 1.96·9-s − 0.974·11-s + 1.19·12-s − 0.731·13-s − 2.48·14-s − 1.21·16-s + 0.233·17-s − 2.56·18-s − 0.612·19-s + 3.29·21-s + 1.26·22-s + 1.21·23-s + 0.682·24-s + 0.952·26-s + 1.67·27-s + 1.33·28-s + 0.396·29-s − 1.20·31-s + 1.18·32-s − 1.67·33-s − 0.304·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212967942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212967942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 7 | \( 1 - 5.05T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 - 0.962T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 - 2.13T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 - 7.77T + 47T^{2} \) |
| 53 | \( 1 - 4.36T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 + 2.79T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 6.10T + 97T^{2} \) |
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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
−8.424680210147907293224915392306, −8.094299886333467686882458249416, −7.45160149034721000017197675885, −7.02455637817515487444199960032, −5.19923009782710586733395488975, −4.72523556016271217821537541847, −3.72924437150669277303495854129, −2.39677470014383545983842617004, −2.09617884377793137427828762393, −1.02380147381149401499960588312,
1.02380147381149401499960588312, 2.09617884377793137427828762393, 2.39677470014383545983842617004, 3.72924437150669277303495854129, 4.72523556016271217821537541847, 5.19923009782710586733395488975, 7.02455637817515487444199960032, 7.45160149034721000017197675885, 8.094299886333467686882458249416, 8.424680210147907293224915392306
