| L(s) = 1 | + 30.4·2-s + 81·3-s + 418.·4-s + 1.05e3·5-s + 2.47e3·6-s − 5.89e3·7-s − 2.86e3·8-s + 6.56e3·9-s + 3.20e4·10-s + 2.63e4·11-s + 3.38e4·12-s − 7.11e4·13-s − 1.79e5·14-s + 8.51e4·15-s − 3.01e5·16-s − 6.60e5·17-s + 2.00e5·18-s − 7.13e5·19-s + 4.39e5·20-s − 4.77e5·21-s + 8.04e5·22-s + 1.72e6·23-s − 2.31e5·24-s − 8.48e5·25-s − 2.17e6·26-s + 5.31e5·27-s − 2.46e6·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.816·4-s + 0.751·5-s + 0.778·6-s − 0.927·7-s − 0.247·8-s + 0.333·9-s + 1.01·10-s + 0.543·11-s + 0.471·12-s − 0.691·13-s − 1.24·14-s + 0.434·15-s − 1.14·16-s − 1.91·17-s + 0.449·18-s − 1.25·19-s + 0.614·20-s − 0.535·21-s + 0.732·22-s + 1.28·23-s − 0.142·24-s − 0.434·25-s − 0.931·26-s + 0.192·27-s − 0.757·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 30.4T + 512T^{2} \) |
| 5 | \( 1 - 1.05e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.89e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.63e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.11e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.60e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.13e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.72e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 8.82e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.38e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.60e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.63e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.25e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.47e6T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.29e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.44e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.12e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.07e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.11e9T + 7.60e17T^{2} \) |
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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
−10.57010821814302143060062219028, −9.405261459674816257531984613683, −8.737423433217838931212422129051, −6.78273648662214770460204681592, −6.38137841564374655769851064108, −4.97283576602514331978895898898, −4.04139646842488964683699978615, −2.87579503202539533591865381704, −2.04047201753696365412129847984, 0,
2.04047201753696365412129847984, 2.87579503202539533591865381704, 4.04139646842488964683699978615, 4.97283576602514331978895898898, 6.38137841564374655769851064108, 6.78273648662214770460204681592, 8.737423433217838931212422129051, 9.405261459674816257531984613683, 10.57010821814302143060062219028
