L(s) = 1 | + 1.70·2-s + 3.27·3-s + 0.900·4-s − 0.246·5-s + 5.57·6-s − 1.87·8-s + 7.73·9-s − 0.420·10-s + 11-s + 2.94·12-s − 3.17·13-s − 0.808·15-s − 4.99·16-s − 6.49·17-s + 13.1·18-s − 4.32·19-s − 0.222·20-s + 1.70·22-s + 3.15·23-s − 6.13·24-s − 4.93·25-s − 5.39·26-s + 15.5·27-s + 6.48·29-s − 1.37·30-s + 1.78·31-s − 4.75·32-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 1.89·3-s + 0.450·4-s − 0.110·5-s + 2.27·6-s − 0.662·8-s + 2.57·9-s − 0.132·10-s + 0.301·11-s + 0.851·12-s − 0.879·13-s − 0.208·15-s − 1.24·16-s − 1.57·17-s + 3.10·18-s − 0.991·19-s − 0.0496·20-s + 0.363·22-s + 0.658·23-s − 1.25·24-s − 0.987·25-s − 1.05·26-s + 2.98·27-s + 1.20·29-s − 0.251·30-s + 0.319·31-s − 0.839·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.076175204\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.076175204\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 + 0.246T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 + 0.553T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 + 3.38T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 0.345T + 71T^{2} \) |
| 73 | \( 1 - 2.97T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−10.85798783745653267022950420415, −9.618797168449614647019501227360, −9.029212376440735187470832952721, −8.238709907993636249546428549350, −7.19192329850018645579938950730, −6.28302916455335774509456014647, −4.52572319182137609865405750995, −4.23740558349972033767329617758, −2.95366520781652709251312131442, −2.24655082177986904060532131087,
2.24655082177986904060532131087, 2.95366520781652709251312131442, 4.23740558349972033767329617758, 4.52572319182137609865405750995, 6.28302916455335774509456014647, 7.19192329850018645579938950730, 8.238709907993636249546428549350, 9.029212376440735187470832952721, 9.618797168449614647019501227360, 10.85798783745653267022950420415