| L(s) = 1 | − 2-s − 1.34·3-s + 4-s + 0.146·5-s + 1.34·6-s + 4.48·7-s − 8-s − 1.20·9-s − 0.146·10-s − 2.99·11-s − 1.34·12-s − 1.07·13-s − 4.48·14-s − 0.196·15-s + 16-s + 1.15·17-s + 1.20·18-s − 2.82·19-s + 0.146·20-s − 6.01·21-s + 2.99·22-s + 2.39·23-s + 1.34·24-s − 4.97·25-s + 1.07·26-s + 5.63·27-s + 4.48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.773·3-s + 0.5·4-s + 0.0656·5-s + 0.547·6-s + 1.69·7-s − 0.353·8-s − 0.401·9-s − 0.0464·10-s − 0.902·11-s − 0.386·12-s − 0.298·13-s − 1.19·14-s − 0.0508·15-s + 0.250·16-s + 0.281·17-s + 0.283·18-s − 0.648·19-s + 0.0328·20-s − 1.31·21-s + 0.637·22-s + 0.500·23-s + 0.273·24-s − 0.995·25-s + 0.211·26-s + 1.08·27-s + 0.847·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 0.146T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 + 0.101T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 9.57T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
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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
−7.83492974968091325345623106493, −7.18062881255668881059039933099, −6.29065955647944963365762274617, −5.51295640230113261841015643210, −5.03768901583744337166976562605, −4.30733791538239435842207334512, −2.93593306576587756763265651577, −2.12045707719048109403565593702, −1.18290904848700247196625327638, 0,
1.18290904848700247196625327638, 2.12045707719048109403565593702, 2.93593306576587756763265651577, 4.30733791538239435842207334512, 5.03768901583744337166976562605, 5.51295640230113261841015643210, 6.29065955647944963365762274617, 7.18062881255668881059039933099, 7.83492974968091325345623106493
