| L(s) = 1 | − 1.41i·5-s + 4.24·7-s + 6i·11-s − 5.65i·13-s + 6·17-s + 4i·19-s + 2.82·23-s + 2.99·25-s + 1.41i·29-s − 1.41·31-s − 6i·35-s − 8.48i·37-s − 2·41-s − 2.82·47-s + 10.9·49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s + 1.60·7-s + 1.80i·11-s − 1.56i·13-s + 1.45·17-s + 0.917i·19-s + 0.589·23-s + 0.599·25-s + 0.262i·29-s − 0.254·31-s − 1.01i·35-s − 1.39i·37-s − 0.312·41-s − 0.412·47-s + 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.653720403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653720403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.238764975319060893677709172928, −7.47807800600067167714553085058, −7.37597063931569378545289341158, −5.81055201962321909278156726938, −5.27641961688018267625040443716, −4.75639865458842942272681861326, −3.95780391678372101158386604980, −2.80867976552438876899849990162, −1.69893721020491212650260759843, −1.06278582659192172698536097450,
0.927739180429062458214564534682, 1.85310805720550208747268603063, 2.97741057396446536165628289629, 3.65326968544984635373047986344, 4.78763101039229189625444828673, 5.20941356756881652149978111763, 6.24929442075676344141022712884, 6.81751759009048057847644059524, 7.70400395548054160211249120098, 8.311509348796417942171612221681
