| L(s) = 1 | + 1.53·2-s + 2.88·3-s + 0.343·4-s + 5-s + 4.42·6-s + 3.86·7-s − 2.53·8-s + 5.33·9-s + 1.53·10-s − 4.75·11-s + 0.992·12-s + 5.91·14-s + 2.88·15-s − 4.56·16-s − 0.640·17-s + 8.16·18-s − 4.70·19-s + 0.343·20-s + 11.1·21-s − 7.27·22-s − 0.308·23-s − 7.32·24-s + 25-s + 6.74·27-s + 1.32·28-s + 2.48·29-s + 4.42·30-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.66·3-s + 0.171·4-s + 0.447·5-s + 1.80·6-s + 1.46·7-s − 0.896·8-s + 1.77·9-s + 0.484·10-s − 1.43·11-s + 0.286·12-s + 1.58·14-s + 0.745·15-s − 1.14·16-s − 0.155·17-s + 1.92·18-s − 1.07·19-s + 0.0768·20-s + 2.43·21-s − 1.55·22-s − 0.0642·23-s − 1.49·24-s + 0.200·25-s + 1.29·27-s + 0.250·28-s + 0.461·29-s + 0.806·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.540913795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.540913795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 2.88T + 3T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 17 | \( 1 + 0.640T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 + 0.308T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 0.635T + 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 - 1.28T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 0.335T + 61T^{2} \) |
| 67 | \( 1 + 0.721T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 0.0141T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−10.17362726549081240163690635410, −9.122253877651024463143136371840, −8.357840221240425626768921274050, −7.965596005847923608831207796417, −6.75756562981011179934032836636, −5.37879737861964838738561772768, −4.74571660089348314215359390024, −3.81274964120908439757957467139, −2.68131201724289554146614173430, −1.99720345050398678253182747149,
1.99720345050398678253182747149, 2.68131201724289554146614173430, 3.81274964120908439757957467139, 4.74571660089348314215359390024, 5.37879737861964838738561772768, 6.75756562981011179934032836636, 7.965596005847923608831207796417, 8.357840221240425626768921274050, 9.122253877651024463143136371840, 10.17362726549081240163690635410
