| L(s) = 1 | − 2-s + 2.35·3-s + 4-s + 0.229·5-s − 2.35·6-s + 7-s − 8-s + 2.52·9-s − 0.229·10-s − 1.24·11-s + 2.35·12-s − 0.120·13-s − 14-s + 0.540·15-s + 16-s − 7.35·17-s − 2.52·18-s − 1.15·19-s + 0.229·20-s + 2.35·21-s + 1.24·22-s + 3.17·23-s − 2.35·24-s − 4.94·25-s + 0.120·26-s − 1.12·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s + 0.102·5-s − 0.959·6-s + 0.377·7-s − 0.353·8-s + 0.841·9-s − 0.0726·10-s − 0.375·11-s + 0.678·12-s − 0.0333·13-s − 0.267·14-s + 0.139·15-s + 0.250·16-s − 1.78·17-s − 0.594·18-s − 0.265·19-s + 0.0514·20-s + 0.512·21-s + 0.265·22-s + 0.662·23-s − 0.479·24-s − 0.989·25-s + 0.0235·26-s − 0.215·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 0.229T + 5T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 + 0.120T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 6.75T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + 3.10T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 - 3.45T + 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.029862985295612910408809295315, −7.12810863309590831935529896900, −6.70503178498030103766160609171, −5.62404199406732295241303120766, −4.70248526894076221164690778991, −3.85809632042073228912643703690, −2.96830289126934672231789779758, −2.24697209899183919588727889141, −1.62427721189776468698863869699, 0,
1.62427721189776468698863869699, 2.24697209899183919588727889141, 2.96830289126934672231789779758, 3.85809632042073228912643703690, 4.70248526894076221164690778991, 5.62404199406732295241303120766, 6.70503178498030103766160609171, 7.12810863309590831935529896900, 8.029862985295612910408809295315
