| L(s) = 1 | + 1.43·2-s + 0.561·3-s − 5.93·4-s + 5·5-s + 0.807·6-s + 31.0·7-s − 20.0·8-s − 26.6·9-s + 7.19·10-s − 3.33·12-s + 45.6·13-s + 44.6·14-s + 2.80·15-s + 18.6·16-s + 40.4·17-s − 38.3·18-s − 91.2·19-s − 29.6·20-s + 17.4·21-s + 32.2·23-s − 11.2·24-s + 25·25-s + 65.6·26-s − 30.1·27-s − 184.·28-s − 35.8·29-s + 4.03·30-s + ⋯ |
| L(s) = 1 | + 0.508·2-s + 0.108·3-s − 0.741·4-s + 0.447·5-s + 0.0549·6-s + 1.67·7-s − 0.885·8-s − 0.988·9-s + 0.227·10-s − 0.0801·12-s + 0.973·13-s + 0.852·14-s + 0.0483·15-s + 0.290·16-s + 0.577·17-s − 0.502·18-s − 1.10·19-s − 0.331·20-s + 0.181·21-s + 0.292·23-s − 0.0957·24-s + 0.200·25-s + 0.494·26-s − 0.214·27-s − 1.24·28-s − 0.229·29-s + 0.0245·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.736868841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736868841\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.43T + 8T^{2} \) |
| 3 | \( 1 - 0.561T + 27T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 13 | \( 1 - 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 230.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 368.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 667.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 84.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 411.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 835.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 799.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 768.T + 9.12e5T^{2} \) |
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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
−10.40335165362817584604611235140, −9.113300283836949239862552560290, −8.502349391220599004124457374645, −7.932210956650347655102640537734, −6.30553913608353578590922633673, −5.49588942808232872576462206606, −4.74930780343235686892296600621, −3.73946661765320663390552653761, −2.38676096773606048045772383350, −0.961052795209085454789672262332,
0.961052795209085454789672262332, 2.38676096773606048045772383350, 3.73946661765320663390552653761, 4.74930780343235686892296600621, 5.49588942808232872576462206606, 6.30553913608353578590922633673, 7.932210956650347655102640537734, 8.502349391220599004124457374645, 9.113300283836949239862552560290, 10.40335165362817584604611235140
