L(s) = 1 | + 2-s − 1.46·3-s + 4-s + 1.82·5-s − 1.46·6-s − 2.77·7-s + 8-s − 0.865·9-s + 1.82·10-s − 1.21·11-s − 1.46·12-s + 1.42·13-s − 2.77·14-s − 2.66·15-s + 16-s + 4.88·17-s − 0.865·18-s + 19-s + 1.82·20-s + 4.05·21-s − 1.21·22-s + 7.05·23-s − 1.46·24-s − 1.67·25-s + 1.42·26-s + 5.64·27-s − 2.77·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.843·3-s + 0.5·4-s + 0.815·5-s − 0.596·6-s − 1.04·7-s + 0.353·8-s − 0.288·9-s + 0.576·10-s − 0.366·11-s − 0.421·12-s + 0.396·13-s − 0.742·14-s − 0.688·15-s + 0.250·16-s + 1.18·17-s − 0.203·18-s + 0.229·19-s + 0.407·20-s + 0.885·21-s − 0.259·22-s + 1.47·23-s − 0.298·24-s − 0.334·25-s + 0.280·26-s + 1.08·27-s − 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 + 7.54T + 37T^{2} \) |
| 41 | \( 1 + 0.862T + 41T^{2} \) |
| 43 | \( 1 + 5.82T + 43T^{2} \) |
| 47 | \( 1 - 8.09T + 47T^{2} \) |
| 53 | \( 1 - 2.56T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 0.424T + 71T^{2} \) |
| 73 | \( 1 - 8.16T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 6.13T + 89T^{2} \) |
| 97 | \( 1 + 9.03T + 97T^{2} \) |
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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
−7.11909062265237472737488745781, −6.67023560548619878006462464527, −5.79950437880825317983857026764, −5.51463856005909147206526861590, −5.08630141859853908719772313158, −3.69678259339430136670911504639, −3.31912383660940084893253005660, −2.34813340585145446113026350942, −1.30359148376382467492414265007, 0,
1.30359148376382467492414265007, 2.34813340585145446113026350942, 3.31912383660940084893253005660, 3.69678259339430136670911504639, 5.08630141859853908719772313158, 5.51463856005909147206526861590, 5.79950437880825317983857026764, 6.67023560548619878006462464527, 7.11909062265237472737488745781