| L(s) = 1 | + 28.5·2-s − 173.·3-s + 300.·4-s + 1.46e3·5-s − 4.93e3·6-s − 2.70e3·7-s − 6.03e3·8-s + 1.03e4·9-s + 4.16e4·10-s − 4.51e4·11-s − 5.20e4·12-s − 5.78e4·13-s − 7.71e4·14-s − 2.53e5·15-s − 3.25e5·16-s + 2.94e5·18-s + 3.23e5·19-s + 4.39e5·20-s + 4.68e5·21-s − 1.28e6·22-s − 2.17e6·23-s + 1.04e6·24-s + 1.86e5·25-s − 1.65e6·26-s + 1.62e6·27-s − 8.12e5·28-s + 2.98e6·29-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 1.23·3-s + 0.586·4-s + 1.04·5-s − 1.55·6-s − 0.425·7-s − 0.520·8-s + 0.524·9-s + 1.31·10-s − 0.930·11-s − 0.724·12-s − 0.562·13-s − 0.536·14-s − 1.29·15-s − 1.24·16-s + 0.660·18-s + 0.570·19-s + 0.614·20-s + 0.525·21-s − 1.17·22-s − 1.62·23-s + 0.642·24-s + 0.0956·25-s − 0.708·26-s + 0.587·27-s − 0.249·28-s + 0.783·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.749736226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749736226\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 28.5T + 512T^{2} \) |
| 3 | \( 1 + 173.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.70e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.51e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.78e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 3.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.28e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.91e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.66e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.04e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.36e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.61e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.95e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.78e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.76e7T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.17188275989104237245810200838, −9.822455417166884339064048644725, −8.200485736460265049325051304006, −6.57185028247457666557613331180, −6.10599896418975791743927556242, −5.24789977056349132361324712240, −4.68290845551813631513357124131, −3.17396912561763064279487831137, −2.15119237079576888953535187381, −0.49014341962645575189135711977,
0.49014341962645575189135711977, 2.15119237079576888953535187381, 3.17396912561763064279487831137, 4.68290845551813631513357124131, 5.24789977056349132361324712240, 6.10599896418975791743927556242, 6.57185028247457666557613331180, 8.200485736460265049325051304006, 9.822455417166884339064048644725, 10.17188275989104237245810200838
