| L(s) = 1 | + 1.13·2-s + 23.9·3-s − 30.7·4-s − 25·5-s + 27.2·6-s − 130.·7-s − 71.3·8-s + 329.·9-s − 28.4·10-s − 121·11-s − 734.·12-s − 1.04e3·13-s − 148.·14-s − 598.·15-s + 901.·16-s + 991.·17-s + 375.·18-s + 361·19-s + 767.·20-s − 3.11e3·21-s − 137.·22-s + 1.54e3·23-s − 1.70e3·24-s + 625·25-s − 1.18e3·26-s + 2.07e3·27-s + 4.00e3·28-s + ⋯ |
| L(s) = 1 | + 0.201·2-s + 1.53·3-s − 0.959·4-s − 0.447·5-s + 0.308·6-s − 1.00·7-s − 0.393·8-s + 1.35·9-s − 0.0899·10-s − 0.301·11-s − 1.47·12-s − 1.70·13-s − 0.202·14-s − 0.686·15-s + 0.880·16-s + 0.832·17-s + 0.272·18-s + 0.229·19-s + 0.429·20-s − 1.54·21-s − 0.0606·22-s + 0.609·23-s − 0.604·24-s + 0.200·25-s − 0.343·26-s + 0.548·27-s + 0.964·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.653756340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653756340\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 1.13T + 32T^{2} \) |
| 3 | \( 1 - 23.9T + 243T^{2} \) |
| 7 | \( 1 + 130.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 991.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.98e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.58e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.27e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.12e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.53e4T + 8.58e9T^{2} \) |
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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
−9.179619357333703327943103000553, −8.478762857581399993662726648542, −7.59908323334160456270654078965, −7.07432519037040117492759335230, −5.53703794532007821153640604447, −4.69296931418581374490566205641, −3.51642121597650399832243280657, −3.24673642211687873131066202192, −2.11846432738187000667294125787, −0.47758213142494842759621590901,
0.47758213142494842759621590901, 2.11846432738187000667294125787, 3.24673642211687873131066202192, 3.51642121597650399832243280657, 4.69296931418581374490566205641, 5.53703794532007821153640604447, 7.07432519037040117492759335230, 7.59908323334160456270654078965, 8.478762857581399993662726648542, 9.179619357333703327943103000553
