| L(s) = 1 | − 2-s − 2.09·3-s + 4-s + 2.51·5-s + 2.09·6-s − 0.00923·7-s − 8-s + 1.37·9-s − 2.51·10-s + 4.67·11-s − 2.09·12-s − 2.67·13-s + 0.00923·14-s − 5.26·15-s + 16-s − 3.35·17-s − 1.37·18-s − 4.27·19-s + 2.51·20-s + 0.0193·21-s − 4.67·22-s − 23-s + 2.09·24-s + 1.34·25-s + 2.67·26-s + 3.40·27-s − 0.00923·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.20·3-s + 0.5·4-s + 1.12·5-s + 0.853·6-s − 0.00348·7-s − 0.353·8-s + 0.457·9-s − 0.796·10-s + 1.40·11-s − 0.603·12-s − 0.741·13-s + 0.00246·14-s − 1.36·15-s + 0.250·16-s − 0.813·17-s − 0.323·18-s − 0.981·19-s + 0.563·20-s + 0.00421·21-s − 0.996·22-s − 0.208·23-s + 0.426·24-s + 0.269·25-s + 0.524·26-s + 0.654·27-s − 0.00174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9327893070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9327893070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 + 0.00923T + 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 + 8.91T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 8.65T + 97T^{2} \) |
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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
−8.350418310120198938652207548027, −6.95056230101383651545154423698, −6.58290691331546933298270713199, −6.26078459248374130735926991843, −5.28657786260209829941833679167, −4.76168912807246933519339158520, −3.66158429117412981110229724107, −2.36613076559894293418489773216, −1.70073955418776468704977554139, −0.60369976558242280908908021559,
0.60369976558242280908908021559, 1.70073955418776468704977554139, 2.36613076559894293418489773216, 3.66158429117412981110229724107, 4.76168912807246933519339158520, 5.28657786260209829941833679167, 6.26078459248374130735926991843, 6.58290691331546933298270713199, 6.95056230101383651545154423698, 8.350418310120198938652207548027
