| L(s) = 1 | + 2-s − 2.42·3-s + 4-s − 1.30·5-s − 2.42·6-s + 3.90·7-s + 8-s + 2.86·9-s − 1.30·10-s + 2.46·11-s − 2.42·12-s − 2.99·13-s + 3.90·14-s + 3.16·15-s + 16-s − 1.93·17-s + 2.86·18-s − 2.79·19-s − 1.30·20-s − 9.45·21-s + 2.46·22-s + 6.30·23-s − 2.42·24-s − 3.28·25-s − 2.99·26-s + 0.333·27-s + 3.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.585·5-s − 0.988·6-s + 1.47·7-s + 0.353·8-s + 0.954·9-s − 0.413·10-s + 0.743·11-s − 0.698·12-s − 0.829·13-s + 1.04·14-s + 0.818·15-s + 0.250·16-s − 0.469·17-s + 0.674·18-s − 0.641·19-s − 0.292·20-s − 2.06·21-s + 0.525·22-s + 1.31·23-s − 0.494·24-s − 0.657·25-s − 0.586·26-s + 0.0642·27-s + 0.738·28-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 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 9.36T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
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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
−7.44270403835160732612333258133, −6.93080081977151657500387275647, −6.20841401302669952331322813682, −5.31885557732211772233792622950, −4.94401634581381834952773249370, −4.34494680167988794842366466669, −3.55425721895902812878305328380, −2.17695636602670688491641259347, −1.33886379059970983213458524079, 0,
1.33886379059970983213458524079, 2.17695636602670688491641259347, 3.55425721895902812878305328380, 4.34494680167988794842366466669, 4.94401634581381834952773249370, 5.31885557732211772233792622950, 6.20841401302669952331322813682, 6.93080081977151657500387275647, 7.44270403835160732612333258133
