| L(s) = 1 | − 2.61·2-s + 4.85·4-s + 5-s − 7-s − 7.47·8-s − 2.61·10-s + 4.23·13-s + 2.61·14-s + 9.85·16-s + 2.47·17-s + 5.47·19-s + 4.85·20-s − 0.472·23-s − 4·25-s − 11.0·26-s − 4.85·28-s − 6.23·29-s + 8.47·31-s − 10.8·32-s − 6.47·34-s − 35-s + 3.47·37-s − 14.3·38-s − 7.47·40-s + 4.47·41-s + 4·43-s + 1.23·46-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s + 0.447·5-s − 0.377·7-s − 2.64·8-s − 0.827·10-s + 1.17·13-s + 0.699·14-s + 2.46·16-s + 0.599·17-s + 1.25·19-s + 1.08·20-s − 0.0984·23-s − 0.800·25-s − 2.17·26-s − 0.917·28-s − 1.15·29-s + 1.52·31-s − 1.91·32-s − 1.10·34-s − 0.169·35-s + 0.570·37-s − 2.32·38-s − 1.18·40-s + 0.698·41-s + 0.609·43-s + 0.182·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027936617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027936617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.904965801330110946848607183565, −7.47561577045377581762215693499, −6.70382492671431974127035999018, −5.93101041532867661772299164012, −5.61136172808665014301371687997, −4.08913388499634500596246846280, −3.16813340021828198557317052522, −2.39998067071151164606138676194, −1.41020087113041439000108157070, −0.73781535584866070058013628589,
0.73781535584866070058013628589, 1.41020087113041439000108157070, 2.39998067071151164606138676194, 3.16813340021828198557317052522, 4.08913388499634500596246846280, 5.61136172808665014301371687997, 5.93101041532867661772299164012, 6.70382492671431974127035999018, 7.47561577045377581762215693499, 7.904965801330110946848607183565
