| L(s) = 1 | − 0.0561·2-s − 1.99·4-s + 2.87·5-s + 3.53·7-s + 0.224·8-s − 0.161·10-s + 11-s + 1.66·13-s − 0.198·14-s + 3.98·16-s + 6.51·17-s − 4.27·19-s − 5.73·20-s − 0.0561·22-s + 8.02·23-s + 3.23·25-s − 0.0934·26-s − 7.06·28-s − 3.26·29-s + 6.97·31-s − 0.672·32-s − 0.365·34-s + 10.1·35-s + 0.826·37-s + 0.239·38-s + 0.644·40-s + 11.9·41-s + ⋯ |
| L(s) = 1 | − 0.0397·2-s − 0.998·4-s + 1.28·5-s + 1.33·7-s + 0.0793·8-s − 0.0509·10-s + 0.301·11-s + 0.461·13-s − 0.0531·14-s + 0.995·16-s + 1.57·17-s − 0.979·19-s − 1.28·20-s − 0.0119·22-s + 1.67·23-s + 0.647·25-s − 0.0183·26-s − 1.33·28-s − 0.606·29-s + 1.25·31-s − 0.118·32-s − 0.0627·34-s + 1.71·35-s + 0.135·37-s + 0.0389·38-s + 0.101·40-s + 1.86·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.847070669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.847070669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0561T + 2T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 0.826T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 - 2.54T + 73T^{2} \) |
| 79 | \( 1 + 6.81T + 79T^{2} \) |
| 83 | \( 1 + 0.562T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.197050099730088802048466013504, −7.58728014338885390873268781250, −6.47564336837382363771111181268, −5.81213088060360925104757218525, −5.13290788795829710764244355459, −4.67915976475497539596388366483, −3.74011365897333462013844152687, −2.72597386113074499817000034813, −1.57024127347974521585597910517, −1.04218039491255058316551189340,
1.04218039491255058316551189340, 1.57024127347974521585597910517, 2.72597386113074499817000034813, 3.74011365897333462013844152687, 4.67915976475497539596388366483, 5.13290788795829710764244355459, 5.81213088060360925104757218525, 6.47564336837382363771111181268, 7.58728014338885390873268781250, 8.197050099730088802048466013504
