| L(s) = 1 | + 2.09e6·2-s + 1.16e10·3-s + 4.39e12·4-s + 7.04e14·5-s + 2.43e16·6-s + 5.58e17·7-s + 9.22e18·8-s − 1.93e20·9-s + 1.47e21·10-s − 1.85e22·11-s + 5.11e22·12-s + 1.59e24·13-s + 1.17e24·14-s + 8.19e24·15-s + 1.93e25·16-s − 1.59e26·17-s − 4.04e26·18-s − 1.91e27·19-s + 3.09e27·20-s + 6.49e27·21-s − 3.89e28·22-s + 2.57e29·23-s + 1.07e29·24-s − 6.40e29·25-s + 3.34e30·26-s − 6.06e30·27-s + 2.45e30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.641·3-s + 0.5·4-s + 0.660·5-s + 0.453·6-s + 0.377·7-s + 0.353·8-s − 0.588·9-s + 0.467·10-s − 0.755·11-s + 0.320·12-s + 1.78·13-s + 0.267·14-s + 0.424·15-s + 0.250·16-s − 0.560·17-s − 0.415·18-s − 0.615·19-s + 0.330·20-s + 0.242·21-s − 0.534·22-s + 1.35·23-s + 0.226·24-s − 0.563·25-s + 1.26·26-s − 1.01·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(5.951389655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.951389655\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.09e6T \) |
| 7 | \( 1 - 5.58e17T \) |
| good | 3 | \( 1 - 1.16e10T + 3.28e20T^{2} \) |
| 5 | \( 1 - 7.04e14T + 1.13e30T^{2} \) |
| 11 | \( 1 + 1.85e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 1.59e24T + 7.93e47T^{2} \) |
| 17 | \( 1 + 1.59e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 1.91e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 2.57e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 2.29e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.28e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 7.27e33T + 2.70e67T^{2} \) |
| 41 | \( 1 - 2.29e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 9.42e34T + 1.73e70T^{2} \) |
| 47 | \( 1 - 6.96e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 1.24e37T + 1.39e74T^{2} \) |
| 59 | \( 1 - 1.64e38T + 1.40e76T^{2} \) |
| 61 | \( 1 + 1.71e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 2.21e39T + 3.32e78T^{2} \) |
| 71 | \( 1 - 6.16e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.87e38T + 1.32e80T^{2} \) |
| 79 | \( 1 + 4.72e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 1.87e41T + 3.31e82T^{2} \) |
| 89 | \( 1 + 9.99e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 9.04e42T + 2.69e85T^{2} \) |
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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
−11.47920899291495745707217439976, −10.53317995008386109296913771473, −8.929854366827503163638041415357, −8.056962467260739984366172823577, −6.44243481745730908514199311639, −5.54246808148052149980814849432, −4.24741978931092285495088923266, −2.99812444587546629751622850212, −2.19506549203257828652961701015, −0.952163731881328260345106802611,
0.952163731881328260345106802611, 2.19506549203257828652961701015, 2.99812444587546629751622850212, 4.24741978931092285495088923266, 5.54246808148052149980814849432, 6.44243481745730908514199311639, 8.056962467260739984366172823577, 8.929854366827503163638041415357, 10.53317995008386109296913771473, 11.47920899291495745707217439976
