| L(s) = 1 | + 324.·2-s + 2.64e4·3-s − 4.19e5·4-s − 5.99e6·5-s + 8.57e6·6-s + 1.98e8·7-s − 3.05e8·8-s − 4.62e8·9-s − 1.94e9·10-s + 1.34e9·11-s − 1.10e10·12-s + 2.69e10·13-s + 6.43e10·14-s − 1.58e11·15-s + 1.20e11·16-s − 6.53e11·17-s − 1.50e11·18-s + 3.36e11·19-s + 2.51e12·20-s + 5.24e12·21-s + 4.36e11·22-s − 5.14e12·23-s − 8.08e12·24-s + 1.68e13·25-s + 8.72e12·26-s − 4.29e13·27-s − 8.31e13·28-s + ⋯ |
| L(s) = 1 | + 0.447·2-s + 0.775·3-s − 0.799·4-s − 1.37·5-s + 0.347·6-s + 1.85·7-s − 0.805·8-s − 0.398·9-s − 0.614·10-s + 0.172·11-s − 0.620·12-s + 0.703·13-s + 0.832·14-s − 1.06·15-s + 0.438·16-s − 1.33·17-s − 0.178·18-s + 0.239·19-s + 1.09·20-s + 1.44·21-s + 0.0770·22-s − 0.595·23-s − 0.625·24-s + 0.885·25-s + 0.315·26-s − 1.08·27-s − 1.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(2.210687058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210687058\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.11e15T \) |
| good | 2 | \( 1 - 324.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 2.64e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 5.99e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.98e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.34e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 2.69e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.53e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 3.36e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 5.14e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.51e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 4.38e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 3.06e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 4.15e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.98e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 1.85e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 8.61e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.37e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.07e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 2.58e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.30e16T + 2.53e35T^{2} \) |
| 79 | \( 1 + 4.21e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.88e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.32e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.89e18T + 5.60e37T^{2} \) |
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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
−11.66832357402219407923241057143, −11.14140002848493480481756738211, −9.002253523955553513479203813018, −8.357666613616467995617333275964, −7.60401793298971387858472316372, −5.51542914013022389172895240616, −4.26701756304925839312287370106, −3.77739047902707551416064648280, −2.18351773755871597129177946686, −0.63863551057649166363476109320,
0.63863551057649166363476109320, 2.18351773755871597129177946686, 3.77739047902707551416064648280, 4.26701756304925839312287370106, 5.51542914013022389172895240616, 7.60401793298971387858472316372, 8.357666613616467995617333275964, 9.002253523955553513479203813018, 11.14140002848493480481756738211, 11.66832357402219407923241057143
