| L(s) = 1 | − 0.561·2-s + 191.·3-s − 511.·4-s + 819.·5-s − 107.·6-s + 1.08e3·7-s + 574.·8-s + 1.70e4·9-s − 460.·10-s − 4.82e4·11-s − 9.80e4·12-s − 1.07e5·13-s − 609.·14-s + 1.57e5·15-s + 2.61e5·16-s − 9.56e3·18-s + 8.72e5·19-s − 4.19e5·20-s + 2.07e5·21-s + 2.71e4·22-s + 1.91e6·23-s + 1.10e5·24-s − 1.28e6·25-s + 6.05e4·26-s − 5.09e5·27-s − 5.55e5·28-s + 4.76e5·29-s + ⋯ |
| L(s) = 1 | − 0.0248·2-s + 1.36·3-s − 0.999·4-s + 0.586·5-s − 0.0338·6-s + 0.170·7-s + 0.0496·8-s + 0.864·9-s − 0.0145·10-s − 0.994·11-s − 1.36·12-s − 1.04·13-s − 0.00424·14-s + 0.800·15-s + 0.998·16-s − 0.0214·18-s + 1.53·19-s − 0.586·20-s + 0.233·21-s + 0.0246·22-s + 1.42·23-s + 0.0677·24-s − 0.655·25-s + 0.0259·26-s − 0.184·27-s − 0.170·28-s + 0.125·29-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 0.561T + 512T^{2} \) |
| 3 | \( 1 - 191.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 819.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.08e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.07e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.72e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.91e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.76e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.17e4T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.41e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.67e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.42e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.59e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.34e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.31e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.30e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.91e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.50e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.40e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.77e8T + 7.60e17T^{2} \) |
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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
−9.674551286803701455714907377459, −8.860150181945039101065579263242, −8.002598529094323454850860680338, −7.26444111171989457623091313824, −5.47027423603937223106119159677, −4.79152845426682876004738698243, −3.38452368687311466140331522189, −2.65276974515352988828508263131, −1.42876497455374686906709236303, 0,
1.42876497455374686906709236303, 2.65276974515352988828508263131, 3.38452368687311466140331522189, 4.79152845426682876004738698243, 5.47027423603937223106119159677, 7.26444111171989457623091313824, 8.002598529094323454850860680338, 8.860150181945039101065579263242, 9.674551286803701455714907377459
