| L(s) = 1 | + 10.8·2-s − 12.9·3-s + 86.2·4-s − 25·5-s − 140.·6-s − 62.2·7-s + 589.·8-s − 76.3·9-s − 271.·10-s − 121·11-s − 1.11e3·12-s − 23.7·13-s − 677.·14-s + 322.·15-s + 3.65e3·16-s + 712.·17-s − 830.·18-s − 361·19-s − 2.15e3·20-s + 804.·21-s − 1.31e3·22-s + 2.86e3·23-s − 7.61e3·24-s + 625·25-s − 258.·26-s + 4.12e3·27-s − 5.37e3·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.828·3-s + 2.69·4-s − 0.447·5-s − 1.59·6-s − 0.480·7-s + 3.25·8-s − 0.314·9-s − 0.859·10-s − 0.301·11-s − 2.23·12-s − 0.0390·13-s − 0.923·14-s + 0.370·15-s + 3.56·16-s + 0.597·17-s − 0.604·18-s − 0.229·19-s − 1.20·20-s + 0.397·21-s − 0.579·22-s + 1.12·23-s − 2.69·24-s + 0.200·25-s − 0.0750·26-s + 1.08·27-s − 1.29·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 10.8T + 32T^{2} \) |
| 3 | \( 1 + 12.9T + 243T^{2} \) |
| 7 | \( 1 + 62.2T + 1.68e4T^{2} \) |
| 13 | \( 1 + 23.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 712.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.23e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.45e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5T + 8.58e9T^{2} \) |
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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
−8.566908646089313412430173532754, −7.41476129134516464046149143865, −6.71196509208786436260288387213, −5.99880454223625573994560150502, −5.18746331655667709762221501533, −4.65988179410227091794789366055, −3.41554253457124322559583866407, −2.94455407814128875958912304154, −1.51113534431506188339079269095, 0,
1.51113534431506188339079269095, 2.94455407814128875958912304154, 3.41554253457124322559583866407, 4.65988179410227091794789366055, 5.18746331655667709762221501533, 5.99880454223625573994560150502, 6.71196509208786436260288387213, 7.41476129134516464046149143865, 8.566908646089313412430173532754
