| L(s) = 1 | − 3i·3-s + (2 − 11i)5-s − 10i·7-s − 9·9-s − 14·11-s − 82i·13-s + (−33 − 6i)15-s + 18i·17-s + 136·19-s − 30·21-s − 140i·23-s + (−117 − 44i)25-s + 27i·27-s + 112·29-s − 72·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.178 − 0.983i)5-s − 0.539i·7-s − 0.333·9-s − 0.383·11-s − 1.74i·13-s + (−0.568 − 0.103i)15-s + 0.256i·17-s + 1.64·19-s − 0.311·21-s − 1.26i·23-s + (−0.936 − 0.351i)25-s + 0.192i·27-s + 0.717·29-s − 0.417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.565205321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565205321\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (-2 + 11i)T \) |
| good | 7 | \( 1 + 10iT - 343T^{2} \) |
| 11 | \( 1 + 14T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 136T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 112T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 26iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 446T + 6.89e4T^{2} \) |
| 43 | \( 1 + 396iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 144iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 158iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 342T + 2.05e5T^{2} \) |
| 61 | \( 1 + 314T + 2.26e5T^{2} \) |
| 67 | \( 1 + 152iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 932T + 3.57e5T^{2} \) |
| 73 | \( 1 - 548iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 512T + 4.93e5T^{2} \) |
| 83 | \( 1 + 284iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 810T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3iT - 9.12e5T^{2} \) |
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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
−9.087178727533971559227174153943, −8.193101576792266532923928447150, −7.72519889518423565352947388724, −6.69660005767685647939162550990, −5.52580361716613793503156274074, −5.07786647093711178099037449032, −3.71594226411493817538148408946, −2.59933203225918230257963052645, −1.16305332793894982902904096909, −0.42539547212711297801773986156,
1.70745926232708190927854686446, 2.86057124007904460095454866060, 3.68706189128463260069871149483, 4.90411268435908300400085592452, 5.72446038286664384309001199709, 6.73533539203123018049264133524, 7.42599115149340842830328351341, 8.544645796890685683780197240594, 9.610778956217537083033905441113, 9.762984975416915095507407077277
