| L(s) = 1 | + 2.62e5·2-s + 5.04e8·3-s + 6.87e10·4-s + 3.81e12·5-s + 1.32e14·6-s − 3.94e15·7-s + 1.80e16·8-s − 1.95e17·9-s + 1.00e18·10-s − 2.18e18·11-s + 3.46e19·12-s + 1.81e20·13-s − 1.03e21·14-s + 1.92e21·15-s + 4.72e21·16-s − 2.91e22·17-s − 5.13e22·18-s − 4.63e23·19-s + 2.62e23·20-s − 1.99e24·21-s − 5.71e23·22-s − 5.39e24·23-s + 9.08e24·24-s + 1.45e25·25-s + 4.76e25·26-s − 3.25e26·27-s − 2.71e26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.751·3-s + 0.5·4-s + 0.447·5-s + 0.531·6-s − 0.916·7-s + 0.353·8-s − 0.435·9-s + 0.316·10-s − 0.118·11-s + 0.375·12-s + 0.447·13-s − 0.647·14-s + 0.336·15-s + 0.250·16-s − 0.503·17-s − 0.307·18-s − 1.02·19-s + 0.223·20-s − 0.688·21-s − 0.0835·22-s − 0.346·23-s + 0.265·24-s + 0.200·25-s + 0.316·26-s − 1.07·27-s − 0.458·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.62e5T \) |
| 5 | \( 1 - 3.81e12T \) |
| good | 3 | \( 1 - 5.04e8T + 4.50e17T^{2} \) |
| 7 | \( 1 + 3.94e15T + 1.85e31T^{2} \) |
| 11 | \( 1 + 2.18e18T + 3.40e38T^{2} \) |
| 13 | \( 1 - 1.81e20T + 1.64e41T^{2} \) |
| 17 | \( 1 + 2.91e22T + 3.36e45T^{2} \) |
| 19 | \( 1 + 4.63e23T + 2.06e47T^{2} \) |
| 23 | \( 1 + 5.39e24T + 2.42e50T^{2} \) |
| 29 | \( 1 + 1.38e27T + 1.28e54T^{2} \) |
| 31 | \( 1 - 1.08e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 8.17e28T + 1.05e58T^{2} \) |
| 41 | \( 1 - 4.55e29T + 4.70e59T^{2} \) |
| 43 | \( 1 + 6.17e29T + 2.74e60T^{2} \) |
| 47 | \( 1 + 1.66e30T + 7.37e61T^{2} \) |
| 53 | \( 1 + 5.11e31T + 6.28e63T^{2} \) |
| 59 | \( 1 + 4.79e32T + 3.32e65T^{2} \) |
| 61 | \( 1 - 1.48e33T + 1.14e66T^{2} \) |
| 67 | \( 1 - 1.20e34T + 3.67e67T^{2} \) |
| 71 | \( 1 + 1.19e32T + 3.13e68T^{2} \) |
| 73 | \( 1 + 4.41e33T + 8.76e68T^{2} \) |
| 79 | \( 1 + 1.73e35T + 1.63e70T^{2} \) |
| 83 | \( 1 + 1.57e35T + 1.01e71T^{2} \) |
| 89 | \( 1 + 9.73e35T + 1.34e72T^{2} \) |
| 97 | \( 1 + 8.76e36T + 3.24e73T^{2} \) |
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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
−12.73660161638334851055762928733, −11.07158561625173846818822523507, −9.605670757377109707356472886777, −8.364178216760477204031401219148, −6.71532226905801905734105790779, −5.65100917576693380863948649946, −3.97178720173430513063504440393, −2.91875255903165886086164931899, −1.90664924263849290447410384716, 0,
1.90664924263849290447410384716, 2.91875255903165886086164931899, 3.97178720173430513063504440393, 5.65100917576693380863948649946, 6.71532226905801905734105790779, 8.364178216760477204031401219148, 9.605670757377109707356472886777, 11.07158561625173846818822523507, 12.73660161638334851055762928733
