| L(s) = 1 | − 1.21·2-s − 0.512·4-s + 5-s + 3.60·7-s + 3.06·8-s − 1.21·10-s − 5.37·11-s − 4.39·14-s − 2.71·16-s + 1.13·17-s + 2.26·19-s − 0.512·20-s + 6.55·22-s + 3.89·23-s + 25-s − 1.84·28-s + 0.0247·29-s + 5.46·31-s − 2.81·32-s − 1.38·34-s + 3.60·35-s − 8.70·37-s − 2.76·38-s + 3.06·40-s − 3.73·41-s − 1.13·43-s + 2.75·44-s + ⋯ |
| L(s) = 1 | − 0.862·2-s − 0.256·4-s + 0.447·5-s + 1.36·7-s + 1.08·8-s − 0.385·10-s − 1.61·11-s − 1.17·14-s − 0.678·16-s + 0.274·17-s + 0.520·19-s − 0.114·20-s + 1.39·22-s + 0.811·23-s + 0.200·25-s − 0.348·28-s + 0.00459·29-s + 0.981·31-s − 0.498·32-s − 0.236·34-s + 0.608·35-s − 1.43·37-s − 0.448·38-s + 0.484·40-s − 0.582·41-s − 0.172·43-s + 0.414·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310935876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310935876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 0.0247T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 0.171T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.996147716252474014066691035762, −7.50332924317211506253640937764, −6.74346071028218048673009468258, −5.49334672492657388579433186380, −5.09134396865681225531407185640, −4.63857679301806920129171115337, −3.44007545219917676071486130728, −2.39736940936600332719783038790, −1.62016166265135182658613918251, −0.68759074507928419909804053108,
0.68759074507928419909804053108, 1.62016166265135182658613918251, 2.39736940936600332719783038790, 3.44007545219917676071486130728, 4.63857679301806920129171115337, 5.09134396865681225531407185640, 5.49334672492657388579433186380, 6.74346071028218048673009468258, 7.50332924317211506253640937764, 7.996147716252474014066691035762
