| L(s) = 1 | − 5.16·2-s + 18.6·4-s + 0.429·5-s − 25.9·7-s − 55.0·8-s − 2.21·10-s − 45.0·11-s + 73.9·13-s + 133.·14-s + 135.·16-s − 57.4·17-s − 5.06·19-s + 8.01·20-s + 232.·22-s + 2.99·23-s − 124.·25-s − 382.·26-s − 483.·28-s − 4.11·29-s + 142.·31-s − 257.·32-s + 296.·34-s − 11.1·35-s − 117.·37-s + 26.1·38-s − 23.6·40-s − 65.2·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s + 0.0384·5-s − 1.39·7-s − 2.43·8-s − 0.0701·10-s − 1.23·11-s + 1.57·13-s + 2.55·14-s + 2.11·16-s − 0.820·17-s − 0.0611·19-s + 0.0896·20-s + 2.25·22-s + 0.0271·23-s − 0.998·25-s − 2.88·26-s − 3.26·28-s − 0.0263·29-s + 0.823·31-s − 1.42·32-s + 1.49·34-s − 0.0537·35-s − 0.523·37-s + 0.111·38-s − 0.0935·40-s − 0.248·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 5.16T + 8T^{2} \) |
| 5 | \( 1 - 0.429T + 125T^{2} \) |
| 7 | \( 1 + 25.9T + 343T^{2} \) |
| 11 | \( 1 + 45.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.06T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.99T + 1.21e4T^{2} \) |
| 29 | \( 1 + 4.11T + 2.43e4T^{2} \) |
| 31 | \( 1 - 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 65.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 558.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 77.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 332.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 461.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 191.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 339.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 610.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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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
−8.503940205776754976309378269047, −7.78526366798865066354098630015, −6.93635605036891286044910967049, −6.29973535932752565017843041738, −5.63896160640086508272806155426, −3.94593693177725463906445407112, −2.92632122475647680829088459524, −2.14425818574549048681084461551, −0.856112220185773169474233025574, 0,
0.856112220185773169474233025574, 2.14425818574549048681084461551, 2.92632122475647680829088459524, 3.94593693177725463906445407112, 5.63896160640086508272806155426, 6.29973535932752565017843041738, 6.93635605036891286044910967049, 7.78526366798865066354098630015, 8.503940205776754976309378269047
