| L(s) = 1 | + 2-s + 1.13·3-s + 4-s − 5-s + 1.13·6-s − 2.24·7-s + 8-s − 1.71·9-s − 10-s + 11-s + 1.13·12-s + 0.202·13-s − 2.24·14-s − 1.13·15-s + 16-s + 2.27·17-s − 1.71·18-s + 6.24·19-s − 20-s − 2.55·21-s + 22-s − 2.51·23-s + 1.13·24-s + 25-s + 0.202·26-s − 5.34·27-s − 2.24·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.655·3-s + 0.5·4-s − 0.447·5-s + 0.463·6-s − 0.850·7-s + 0.353·8-s − 0.570·9-s − 0.316·10-s + 0.301·11-s + 0.327·12-s + 0.0560·13-s − 0.601·14-s − 0.292·15-s + 0.250·16-s + 0.551·17-s − 0.403·18-s + 1.43·19-s − 0.223·20-s − 0.556·21-s + 0.213·22-s − 0.524·23-s + 0.231·24-s + 0.200·25-s + 0.0396·26-s − 1.02·27-s − 0.425·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.316419050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.316419050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 13 | \( 1 - 0.202T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 0.359T + 43T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 - 3.20T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 9.86T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.82048125424607369587594026228, −7.10790024136854848085064947361, −6.42834596651968354860959889887, −5.70696300411982961242521738820, −5.05841398407013140280398015130, −4.07489051272728369552867556013, −3.33974259455065182970960464951, −3.06681595640479053273539167375, −2.06525860984018595557207696037, −0.76991672965328379132018318832,
0.76991672965328379132018318832, 2.06525860984018595557207696037, 3.06681595640479053273539167375, 3.33974259455065182970960464951, 4.07489051272728369552867556013, 5.05841398407013140280398015130, 5.70696300411982961242521738820, 6.42834596651968354860959889887, 7.10790024136854848085064947361, 7.82048125424607369587594026228
