| L(s) = 1 | − 8.19e3·2-s + 5.48e6·3-s + 6.71e7·4-s − 4.49e10·6-s + 8.15e10·7-s − 5.49e11·8-s + 2.25e13·9-s + 1.33e13·11-s + 3.68e14·12-s − 5.64e14·13-s − 6.68e14·14-s + 4.50e15·16-s + 1.28e16·17-s − 1.84e17·18-s + 3.07e17·19-s + 4.47e17·21-s − 1.09e17·22-s + 1.99e18·23-s − 3.01e18·24-s + 4.62e18·26-s + 8.17e19·27-s + 5.47e18·28-s − 8.80e19·29-s + 2.22e20·31-s − 3.68e19·32-s + 7.35e19·33-s − 1.05e20·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.98·3-s + 0.5·4-s − 1.40·6-s + 0.318·7-s − 0.353·8-s + 2.95·9-s + 0.116·11-s + 0.993·12-s − 0.517·13-s − 0.225·14-s + 0.250·16-s + 0.315·17-s − 2.08·18-s + 1.67·19-s + 0.632·21-s − 0.0827·22-s + 0.826·23-s − 0.702·24-s + 0.365·26-s + 3.88·27-s + 0.159·28-s − 1.59·29-s + 1.63·31-s − 0.176·32-s + 0.232·33-s − 0.223·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(4.838573487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.838573487\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.19e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.48e6T + 7.62e12T^{2} \) |
| 7 | \( 1 - 8.15e10T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.33e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 5.64e14T + 1.19e30T^{2} \) |
| 17 | \( 1 - 1.28e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 3.07e17T + 3.36e34T^{2} \) |
| 23 | \( 1 - 1.99e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 8.80e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 2.22e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 1.61e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 9.68e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 9.33e21T + 1.26e44T^{2} \) |
| 47 | \( 1 + 1.20e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 8.75e21T + 3.59e46T^{2} \) |
| 59 | \( 1 - 3.03e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 1.15e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 1.86e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 8.93e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 2.18e25T + 2.04e50T^{2} \) |
| 79 | \( 1 - 1.05e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 2.25e25T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.78e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 8.87e26T + 4.39e53T^{2} \) |
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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
−9.980776586076371331808288903361, −9.503942110243971386433206820636, −8.443569144796550680047038976029, −7.71218886389618433273048877266, −6.91564270823109594260183730863, −4.93116096784803496001811203679, −3.54449076715405071809449859382, −2.80973894543944553299525949270, −1.79172454223024456472182810996, −0.961111471046115933469520989046,
0.961111471046115933469520989046, 1.79172454223024456472182810996, 2.80973894543944553299525949270, 3.54449076715405071809449859382, 4.93116096784803496001811203679, 6.91564270823109594260183730863, 7.71218886389618433273048877266, 8.443569144796550680047038976029, 9.503942110243971386433206820636, 9.980776586076371331808288903361
