| L(s) = 1 | − 2-s + 2.05·3-s + 4-s + 1.96·5-s − 2.05·6-s − 4.12·7-s − 8-s + 1.24·9-s − 1.96·10-s + 2.05·12-s + 4.76·13-s + 4.12·14-s + 4.04·15-s + 16-s − 7.01·17-s − 1.24·18-s + 19-s + 1.96·20-s − 8.49·21-s + 3.10·23-s − 2.05·24-s − 1.13·25-s − 4.76·26-s − 3.62·27-s − 4.12·28-s + 0.675·29-s − 4.04·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.18·3-s + 0.5·4-s + 0.878·5-s − 0.840·6-s − 1.55·7-s − 0.353·8-s + 0.413·9-s − 0.621·10-s + 0.594·12-s + 1.32·13-s + 1.10·14-s + 1.04·15-s + 0.250·16-s − 1.70·17-s − 0.292·18-s + 0.229·19-s + 0.439·20-s − 1.85·21-s + 0.648·23-s − 0.420·24-s − 0.227·25-s − 0.935·26-s − 0.696·27-s − 0.779·28-s + 0.125·29-s − 0.738·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 + 7.01T + 17T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 - 0.675T + 29T^{2} \) |
| 31 | \( 1 + 0.811T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 2.49T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.319977104730521454635767768541, −7.21630187100423510902873169285, −6.52214625892717992563426274300, −6.19189496870367669635284108182, −5.09140005815940778627922835871, −3.63120036365532933898651215252, −3.30425926461307540430937467318, −2.34398505104930655710516025041, −1.62358303113206153404125337297, 0,
1.62358303113206153404125337297, 2.34398505104930655710516025041, 3.30425926461307540430937467318, 3.63120036365532933898651215252, 5.09140005815940778627922835871, 6.19189496870367669635284108182, 6.52214625892717992563426274300, 7.21630187100423510902873169285, 8.319977104730521454635767768541
