L(s) = 1 | − 25.8·2-s + 225.·3-s + 154.·4-s + 96.4·5-s − 5.81e3·6-s − 1.38e3·7-s + 9.22e3·8-s + 3.10e4·9-s − 2.49e3·10-s − 3.38e4·11-s + 3.48e4·12-s + 1.38e5·13-s + 3.57e4·14-s + 2.17e4·15-s − 3.17e5·16-s − 8.00e5·18-s − 4.29e5·19-s + 1.49e4·20-s − 3.12e5·21-s + 8.73e5·22-s + 3.52e5·23-s + 2.07e6·24-s − 1.94e6·25-s − 3.57e6·26-s + 2.54e6·27-s − 2.14e5·28-s − 5.08e5·29-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 1.60·3-s + 0.302·4-s + 0.0690·5-s − 1.83·6-s − 0.218·7-s + 0.796·8-s + 1.57·9-s − 0.0787·10-s − 0.696·11-s + 0.485·12-s + 1.34·13-s + 0.248·14-s + 0.110·15-s − 1.21·16-s − 1.79·18-s − 0.756·19-s + 0.0208·20-s − 0.350·21-s + 0.795·22-s + 0.262·23-s + 1.27·24-s − 0.995·25-s − 1.53·26-s + 0.923·27-s − 0.0659·28-s − 0.133·29-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 | \( 1 + 25.8T + 512T^{2} \) |
| 3 | \( 1 - 225.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 96.4T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.38e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.38e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.38e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.29e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.52e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.08e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.73e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.62e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.12e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.43e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.53e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.44e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.38e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.61e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.04e9T + 7.60e17T^{2} \) |
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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
−9.275180302960141281466122360639, −9.008012101516819221438313190517, −7.944338047752003042769381822862, −7.59107678632131907759435973064, −6.10852316189155076031288152411, −4.38568854561951730435667927614, −3.40848825183125641502571834361, −2.23646306710197895425660814904, −1.37015951443949435443936759767, 0,
1.37015951443949435443936759767, 2.23646306710197895425660814904, 3.40848825183125641502571834361, 4.38568854561951730435667927614, 6.10852316189155076031288152411, 7.59107678632131907759435973064, 7.944338047752003042769381822862, 9.008012101516819221438313190517, 9.275180302960141281466122360639