| L(s) = 1 | − 4.53·2-s + 15.7·3-s − 11.4·4-s + 25·5-s − 71.2·6-s + 196.·8-s + 4.27·9-s − 113.·10-s + 126.·11-s − 180.·12-s − 833.·13-s + 393.·15-s − 526.·16-s − 157.·17-s − 19.3·18-s + 1.00e3·19-s − 286.·20-s − 572.·22-s + 73.5·23-s + 3.09e3·24-s + 625·25-s + 3.77e3·26-s − 3.75e3·27-s − 3.69e3·29-s − 1.78e3·30-s − 4.80e3·31-s − 3.91e3·32-s + ⋯ |
| L(s) = 1 | − 0.801·2-s + 1.00·3-s − 0.357·4-s + 0.447·5-s − 0.808·6-s + 1.08·8-s + 0.0175·9-s − 0.358·10-s + 0.314·11-s − 0.360·12-s − 1.36·13-s + 0.451·15-s − 0.514·16-s − 0.132·17-s − 0.0140·18-s + 0.638·19-s − 0.160·20-s − 0.252·22-s + 0.0289·23-s + 1.09·24-s + 0.200·25-s + 1.09·26-s − 0.991·27-s − 0.815·29-s − 0.361·30-s − 0.897·31-s − 0.676·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.53T + 32T^{2} \) |
| 3 | \( 1 - 15.7T + 243T^{2} \) |
| 11 | \( 1 - 126.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 833.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 157.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 73.5T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 195.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.60e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e5T + 8.58e9T^{2} \) |
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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
−10.32953835407596323011627227337, −9.421801307958267867743040305561, −9.077212811320860229291343398425, −7.927752582290760343592510745946, −7.24287293307003048954079776719, −5.55527340783981627497654046477, −4.29400276424212064239920995444, −2.85359460389390226614597475948, −1.63523023239408814873594031231, 0,
1.63523023239408814873594031231, 2.85359460389390226614597475948, 4.29400276424212064239920995444, 5.55527340783981627497654046477, 7.24287293307003048954079776719, 7.927752582290760343592510745946, 9.077212811320860229291343398425, 9.421801307958267867743040305561, 10.32953835407596323011627227337
