L(s) = 1 | + 2-s − 1.17·3-s + 4-s + 3.48·5-s − 1.17·6-s − 3.41·7-s + 8-s − 1.61·9-s + 3.48·10-s − 0.950·11-s − 1.17·12-s − 2.90·13-s − 3.41·14-s − 4.10·15-s + 16-s − 5.56·17-s − 1.61·18-s − 7.73·19-s + 3.48·20-s + 4.02·21-s − 0.950·22-s − 2.78·23-s − 1.17·24-s + 7.12·25-s − 2.90·26-s + 5.43·27-s − 3.41·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.680·3-s + 0.5·4-s + 1.55·5-s − 0.481·6-s − 1.29·7-s + 0.353·8-s − 0.536·9-s + 1.10·10-s − 0.286·11-s − 0.340·12-s − 0.805·13-s − 0.912·14-s − 1.05·15-s + 0.250·16-s − 1.34·17-s − 0.379·18-s − 1.77·19-s + 0.778·20-s + 0.878·21-s − 0.202·22-s − 0.581·23-s − 0.240·24-s + 1.42·25-s − 0.569·26-s + 1.04·27-s − 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 0.950T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 6.72T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 0.871T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−9.381128534912382988537756332543, −8.373693746455833380613259446603, −6.87464977687083403210734562419, −6.41362713701588361236096272557, −5.86538625682687556586717620371, −5.12909152251538350025946723109, −4.10005832544152378752607133967, −2.68226161216218955950654985262, −2.15334480750620276171900107694, 0,
2.15334480750620276171900107694, 2.68226161216218955950654985262, 4.10005832544152378752607133967, 5.12909152251538350025946723109, 5.86538625682687556586717620371, 6.41362713701588361236096272557, 6.87464977687083403210734562419, 8.373693746455833380613259446603, 9.381128534912382988537756332543