| L(s) = 1 | + (0.866 − 1.5i)3-s + (−1.5 − 2.59i)9-s − 3·11-s + 2i·13-s − 5.19·17-s − 5.19i·19-s − 6i·23-s − 5.19·27-s − 10.3i·29-s − 3.46i·31-s + (−2.59 + 4.5i)33-s + 8i·37-s + (3 + 1.73i)39-s + 5.19i·41-s − 3.46·43-s + ⋯ |
| L(s) = 1 | + (0.499 − 0.866i)3-s + (−0.5 − 0.866i)9-s − 0.904·11-s + 0.554i·13-s − 1.26·17-s − 1.19i·19-s − 1.25i·23-s − 1.00·27-s − 1.92i·29-s − 0.622i·31-s + (−0.452 + 0.783i)33-s + 1.31i·37-s + (0.480 + 0.277i)39-s + 0.811i·41-s − 0.528·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095209602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095209602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 5.19iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.304598724352314735916215781988, −8.367122171834823011825924382111, −7.957994021730337527810777364493, −6.68670469323539407150192765260, −6.51088446839144384115259268595, −5.08948486018141563716444767019, −4.15376894899442039769270520641, −2.74845397359810433691047741210, −2.13322853984231446408569863864, −0.40530332179115754764925378030,
1.93729231933103658293255531009, 3.11254192744972774004589500809, 3.88497527519004272860571672959, 5.06966633744549355466352372895, 5.55994684587194793855413809038, 6.89929429570006138288519963623, 7.86098552329013549100328323186, 8.493754748499875612504475369035, 9.321255531527514194447802277528, 10.07203805406103123377616966038
