| L(s) = 1 | − 5.05·2-s − 6.23·3-s + 17.5·4-s + 18.8·5-s + 31.5·6-s − 48.5·8-s + 11.8·9-s − 95.2·10-s + 7.18·11-s − 109.·12-s + 13·13-s − 117.·15-s + 104.·16-s + 110.·17-s − 59.8·18-s + 103.·19-s + 331.·20-s − 36.3·22-s + 59.7·23-s + 302.·24-s + 229.·25-s − 65.7·26-s + 94.5·27-s − 229.·29-s + 593.·30-s + 252.·31-s − 141.·32-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.19·3-s + 2.19·4-s + 1.68·5-s + 2.14·6-s − 2.14·8-s + 0.438·9-s − 3.01·10-s + 0.196·11-s − 2.63·12-s + 0.277·13-s − 2.01·15-s + 1.63·16-s + 1.58·17-s − 0.783·18-s + 1.25·19-s + 3.70·20-s − 0.352·22-s + 0.541·23-s + 2.57·24-s + 1.83·25-s − 0.496·26-s + 0.673·27-s − 1.46·29-s + 3.61·30-s + 1.46·31-s − 0.781·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8740549139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8740549139\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 5.05T + 8T^{2} \) |
| 3 | \( 1 + 6.23T + 27T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 11 | \( 1 - 7.18T + 1.33e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 59.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 44.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 82.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.58T + 2.26e5T^{2} \) |
| 67 | \( 1 + 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 849.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 184.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 482.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.78e3T + 9.12e5T^{2} \) |
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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
−10.12509973269486517553457554723, −9.514760050878858360612647847523, −8.729901253998313703534664877473, −7.53393954659721328841987235107, −6.66817526982635892158480229725, −5.83016685641442517885654495533, −5.31131335856563119055053092595, −2.92408850674410124758235794130, −1.54954691305521978022400681216, −0.839961680932392632255188834272,
0.839961680932392632255188834272, 1.54954691305521978022400681216, 2.92408850674410124758235794130, 5.31131335856563119055053092595, 5.83016685641442517885654495533, 6.66817526982635892158480229725, 7.53393954659721328841987235107, 8.729901253998313703534664877473, 9.514760050878858360612647847523, 10.12509973269486517553457554723
