L(s) = 1 | − 1.32·2-s − 0.239·4-s − 5-s − 1.05·7-s + 2.97·8-s + 1.32·10-s − 2.49·11-s + 1.40·14-s − 3.46·16-s − 4.95·17-s + 6.35·19-s + 0.239·20-s + 3.30·22-s + 3.62·23-s + 25-s + 0.253·28-s + 0.234·29-s + 1.31·31-s − 1.34·32-s + 6.57·34-s + 1.05·35-s − 6.11·37-s − 8.43·38-s − 2.97·40-s − 8.85·41-s + 9.66·43-s + 0.596·44-s + ⋯ |
L(s) = 1 | − 0.938·2-s − 0.119·4-s − 0.447·5-s − 0.400·7-s + 1.05·8-s + 0.419·10-s − 0.751·11-s + 0.375·14-s − 0.866·16-s − 1.20·17-s + 1.45·19-s + 0.0535·20-s + 0.705·22-s + 0.755·23-s + 0.200·25-s + 0.0478·28-s + 0.0434·29-s + 0.236·31-s − 0.237·32-s + 1.12·34-s + 0.178·35-s − 1.00·37-s − 1.36·38-s − 0.469·40-s − 1.38·41-s + 1.47·43-s + 0.0899·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 0.234T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 + 8.85T + 41T^{2} \) |
| 43 | \( 1 - 9.66T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 - 2.10T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.58403218504945619087310917430, −7.12051930756797616254239515893, −6.38460142446002638494726125863, −5.19723014083484071226654177383, −4.87275257151627480207657620675, −3.85554216487227845258021393530, −3.09350781021702618703344409827, −2.08607632133266992224587384840, −0.947683899150584941891815920893, 0,
0.947683899150584941891815920893, 2.08607632133266992224587384840, 3.09350781021702618703344409827, 3.85554216487227845258021393530, 4.87275257151627480207657620675, 5.19723014083484071226654177383, 6.38460142446002638494726125863, 7.12051930756797616254239515893, 7.58403218504945619087310917430