| L(s) = 1 | + 1.97·2-s − 0.571·3-s + 1.89·4-s + 1.23·5-s − 1.12·6-s − 2.55·7-s − 0.199·8-s − 2.67·9-s + 2.42·10-s − 11-s − 1.08·12-s − 5.05·14-s − 0.703·15-s − 4.19·16-s + 5.97·17-s − 5.27·18-s − 5.32·19-s + 2.33·20-s + 1.46·21-s − 1.97·22-s − 3.26·23-s + 0.113·24-s − 3.48·25-s + 3.24·27-s − 4.85·28-s − 2.86·29-s − 1.38·30-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 0.329·3-s + 0.949·4-s + 0.550·5-s − 0.460·6-s − 0.966·7-s − 0.0704·8-s − 0.891·9-s + 0.768·10-s − 0.301·11-s − 0.313·12-s − 1.34·14-s − 0.181·15-s − 1.04·16-s + 1.44·17-s − 1.24·18-s − 1.22·19-s + 0.522·20-s + 0.319·21-s − 0.420·22-s − 0.680·23-s + 0.0232·24-s − 0.697·25-s + 0.623·27-s − 0.918·28-s − 0.531·29-s − 0.253·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 + 0.571T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 5.32T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 - 0.858T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 8.76T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 7.57T + 79T^{2} \) |
| 83 | \( 1 + 0.707T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−8.897129130455644564574609455201, −7.980849605548338728529923955169, −6.78667632293580570361897136122, −6.07428710550227275860222735440, −5.69384623931574287316867474974, −4.89906835619703709664178531369, −3.74995293763354952055342576403, −3.11554143288322390889109621404, −2.11674258232881231131130643950, 0,
2.11674258232881231131130643950, 3.11554143288322390889109621404, 3.74995293763354952055342576403, 4.89906835619703709664178531369, 5.69384623931574287316867474974, 6.07428710550227275860222735440, 6.78667632293580570361897136122, 7.980849605548338728529923955169, 8.897129130455644564574609455201
