| L(s) = 1 | − 4.78·2-s + 5.72·3-s + 14.9·4-s + 0.431·5-s − 27.4·6-s − 26.0·7-s − 33.0·8-s + 5.79·9-s − 2.06·10-s + 85.3·12-s − 64.5·13-s + 124.·14-s + 2.47·15-s + 39.0·16-s + 17·17-s − 27.7·18-s − 87.1·19-s + 6.43·20-s − 149.·21-s + 197.·23-s − 189.·24-s − 124.·25-s + 308.·26-s − 121.·27-s − 388.·28-s + 154.·29-s − 11.8·30-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.10·3-s + 1.86·4-s + 0.0386·5-s − 1.86·6-s − 1.40·7-s − 1.46·8-s + 0.214·9-s − 0.0653·10-s + 2.05·12-s − 1.37·13-s + 2.38·14-s + 0.0425·15-s + 0.610·16-s + 0.242·17-s − 0.363·18-s − 1.05·19-s + 0.0719·20-s − 1.55·21-s + 1.79·23-s − 1.61·24-s − 0.998·25-s + 2.32·26-s − 0.865·27-s − 2.62·28-s + 0.986·29-s − 0.0720·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5120746758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5120746758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.78T + 8T^{2} \) |
| 3 | \( 1 - 5.72T + 27T^{2} \) |
| 5 | \( 1 - 0.431T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 13 | \( 1 + 64.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 87.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 348.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 567.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 216.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 443.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 814.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 285.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 404.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 414.T + 9.12e5T^{2} \) |
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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
−8.933384200207751101176219832955, −8.218218555673064479076190056319, −7.41975751130346687014668813235, −6.86938897568260172901506260818, −6.05497214039773627167305133410, −4.62094993649123922727964328194, −3.18736948307611827939346301561, −2.74685801636018620124218098988, −1.79357921340309369417994058603, −0.37893092118538678670030615413,
0.37893092118538678670030615413, 1.79357921340309369417994058603, 2.74685801636018620124218098988, 3.18736948307611827939346301561, 4.62094993649123922727964328194, 6.05497214039773627167305133410, 6.86938897568260172901506260818, 7.41975751130346687014668813235, 8.218218555673064479076190056319, 8.933384200207751101176219832955
