| L(s) = 1 | + 1.54·2-s − 2.55·3-s + 0.377·4-s + 5-s − 3.93·6-s − 2.50·8-s + 3.52·9-s + 1.54·10-s + 1.87·11-s − 0.963·12-s + 1.64·13-s − 2.55·15-s − 4.61·16-s + 0.0651·17-s + 5.43·18-s − 7.39·19-s + 0.377·20-s + 2.89·22-s + 2.89·23-s + 6.39·24-s + 25-s + 2.54·26-s − 1.33·27-s + 6.77·29-s − 3.93·30-s − 31-s − 2.10·32-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 1.47·3-s + 0.188·4-s + 0.447·5-s − 1.60·6-s − 0.884·8-s + 1.17·9-s + 0.487·10-s + 0.565·11-s − 0.278·12-s + 0.457·13-s − 0.659·15-s − 1.15·16-s + 0.0157·17-s + 1.28·18-s − 1.69·19-s + 0.0843·20-s + 0.616·22-s + 0.602·23-s + 1.30·24-s + 0.200·25-s + 0.498·26-s − 0.257·27-s + 1.25·29-s − 0.718·30-s − 0.179·31-s − 0.372·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.55T + 3T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 - 0.0651T + 17T^{2} \) |
| 19 | \( 1 + 7.39T + 19T^{2} \) |
| 23 | \( 1 - 2.89T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 2.08T + 71T^{2} \) |
| 73 | \( 1 + 0.990T + 73T^{2} \) |
| 79 | \( 1 - 3.53T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−6.88307706080862239483144721682, −6.59130521844699532942810235572, −6.02321442506258793845067152413, −5.41210880987313250861456013638, −4.77958025903393114961652069723, −4.23748580673328366793730156718, −3.41257286846994732970364674841, −2.34508585594425234922845738208, −1.17395100346559468648988392795, 0,
1.17395100346559468648988392795, 2.34508585594425234922845738208, 3.41257286846994732970364674841, 4.23748580673328366793730156718, 4.77958025903393114961652069723, 5.41210880987313250861456013638, 6.02321442506258793845067152413, 6.59130521844699532942810235572, 6.88307706080862239483144721682
