| L(s) = 1 | − 2.62e5·2-s + 1.16e9·3-s + 6.87e10·4-s − 1.00e13·5-s − 3.04e14·6-s − 3.32e15·7-s − 1.80e16·8-s + 9.00e17·9-s + 2.62e18·10-s − 2.20e19·11-s + 7.98e19·12-s − 2.47e20·13-s + 8.70e20·14-s − 1.16e22·15-s + 4.72e21·16-s + 6.99e22·17-s − 2.36e23·18-s − 8.30e23·19-s − 6.89e23·20-s − 3.85e24·21-s + 5.77e24·22-s − 1.05e25·23-s − 2.09e25·24-s + 2.77e25·25-s + 6.47e25·26-s + 5.23e26·27-s − 2.28e26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 0.5·4-s − 1.17·5-s − 1.22·6-s − 0.770·7-s − 0.353·8-s + 1.99·9-s + 0.831·10-s − 1.19·11-s + 0.865·12-s − 0.609·13-s + 0.544·14-s − 2.03·15-s + 0.250·16-s + 1.20·17-s − 1.41·18-s − 1.83·19-s − 0.587·20-s − 1.33·21-s + 0.844·22-s − 0.678·23-s − 0.612·24-s + 0.382·25-s + 0.430·26-s + 1.73·27-s − 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.62e5T \) |
| good | 3 | \( 1 - 1.16e9T + 4.50e17T^{2} \) |
| 5 | \( 1 + 1.00e13T + 7.27e25T^{2} \) |
| 7 | \( 1 + 3.32e15T + 1.85e31T^{2} \) |
| 11 | \( 1 + 2.20e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 2.47e20T + 1.64e41T^{2} \) |
| 17 | \( 1 - 6.99e22T + 3.36e45T^{2} \) |
| 19 | \( 1 + 8.30e23T + 2.06e47T^{2} \) |
| 23 | \( 1 + 1.05e25T + 2.42e50T^{2} \) |
| 29 | \( 1 + 2.42e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 1.93e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 8.55e28T + 1.05e58T^{2} \) |
| 41 | \( 1 - 5.59e29T + 4.70e59T^{2} \) |
| 43 | \( 1 + 3.15e29T + 2.74e60T^{2} \) |
| 47 | \( 1 - 1.56e30T + 7.37e61T^{2} \) |
| 53 | \( 1 - 3.66e31T + 6.28e63T^{2} \) |
| 59 | \( 1 - 8.04e32T + 3.32e65T^{2} \) |
| 61 | \( 1 - 1.21e33T + 1.14e66T^{2} \) |
| 67 | \( 1 - 1.45e33T + 3.67e67T^{2} \) |
| 71 | \( 1 + 2.91e34T + 3.13e68T^{2} \) |
| 73 | \( 1 - 3.11e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 8.04e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 2.48e35T + 1.01e71T^{2} \) |
| 89 | \( 1 + 1.00e36T + 1.34e72T^{2} \) |
| 97 | \( 1 - 2.19e36T + 3.24e73T^{2} \) |
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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
−18.93681688401047451573014777199, −16.01541736793047335697822960656, −14.84506040264417134492511671913, −12.71627125398104200115109036481, −10.08123335997394365918979508618, −8.422048796732566740039092975693, −7.45753467676015678922558812796, −3.69410166133358014376711419365, −2.39387518847156318666877916962, 0,
2.39387518847156318666877916962, 3.69410166133358014376711419365, 7.45753467676015678922558812796, 8.422048796732566740039092975693, 10.08123335997394365918979508618, 12.71627125398104200115109036481, 14.84506040264417134492511671913, 16.01541736793047335697822960656, 18.93681688401047451573014777199
