| L(s) = 1 | − 0.150·2-s − 1.97·4-s − 1.54·7-s + 0.600·8-s − 4.56·11-s − 1.09·13-s + 0.233·14-s + 3.86·16-s − 17-s + 4.67·19-s + 0.688·22-s − 0.529·23-s + 0.165·26-s + 3.05·28-s − 8.06·29-s − 4.78·31-s − 1.78·32-s + 0.150·34-s − 5.27·37-s − 0.705·38-s + 0.751·41-s + 9.49·43-s + 9.01·44-s + 0.0799·46-s − 10.7·47-s − 4.61·49-s + 2.17·52-s + ⋯ |
| L(s) = 1 | − 0.106·2-s − 0.988·4-s − 0.583·7-s + 0.212·8-s − 1.37·11-s − 0.304·13-s + 0.0623·14-s + 0.965·16-s − 0.242·17-s + 1.07·19-s + 0.146·22-s − 0.110·23-s + 0.0324·26-s + 0.577·28-s − 1.49·29-s − 0.858·31-s − 0.315·32-s + 0.0258·34-s − 0.867·37-s − 0.114·38-s + 0.117·41-s + 1.44·43-s + 1.35·44-s + 0.0117·46-s − 1.56·47-s − 0.659·49-s + 0.300·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6806630196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6806630196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.150T + 2T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.529T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 - 0.751T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0227T + 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 9.75T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 6.78T + 97T^{2} \) |
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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
−8.512887093997554318713704144909, −7.71761638078614755507916925038, −7.30732479924391044274521203592, −6.12895638654820996041064416321, −5.32335573018455825172372076418, −4.91097216616634626552248134008, −3.77054402803082828816321279934, −3.14880218264278577406321386117, −1.98263318586716852321741456892, −0.47101239605874939706575889770,
0.47101239605874939706575889770, 1.98263318586716852321741456892, 3.14880218264278577406321386117, 3.77054402803082828816321279934, 4.91097216616634626552248134008, 5.32335573018455825172372076418, 6.12895638654820996041064416321, 7.30732479924391044274521203592, 7.71761638078614755507916925038, 8.512887093997554318713704144909
