L(s) = 1 | + 2.84i·3-s + (0.691 − 2.12i)5-s − 0.145i·7-s − 5.10·9-s + 5.71·11-s + 5.24i·13-s + (6.05 + 1.96i)15-s − 7.15i·17-s + 19-s + 0.412·21-s + 0.622i·23-s + (−4.04 − 2.94i)25-s − 6.00i·27-s + 5.46·29-s + 3.77·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + (0.309 − 0.950i)5-s − 0.0548i·7-s − 1.70·9-s + 1.72·11-s + 1.45i·13-s + (1.56 + 0.508i)15-s − 1.73i·17-s + 0.229·19-s + 0.0901·21-s + 0.129i·23-s + (−0.808 − 0.588i)25-s − 1.15i·27-s + 1.01·29-s + 0.678·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.962373912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.962373912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.691 + 2.12i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.84iT - 3T^{2} \) |
| 7 | \( 1 + 0.145iT - 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 - 5.24iT - 13T^{2} \) |
| 17 | \( 1 + 7.15iT - 17T^{2} \) |
| 23 | \( 1 - 0.622iT - 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 - 5.03iT - 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 5.85iT - 47T^{2} \) |
| 53 | \( 1 + 6.97iT - 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 + 8.16T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 - 7.44iT - 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 - 2.17iT - 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 + 4.98iT - 97T^{2} \) |
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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
−9.688145478827945684007999974394, −8.994478917215631643926262168220, −8.587509997347669803833019339851, −7.09949181505727591634125011624, −6.22497232793593668368454619660, −5.20191899164261512619694374484, −4.39913364856039961940055294990, −4.11978257236709199208711908587, −2.79602950245416530076975536493, −1.18480890710461835710753949727,
0.975240058292870222364100414435, 1.96345648673209389123102334946, 2.98443584051759968539176604089, 3.96817626991333269538831862234, 5.72846058891521949810104398215, 6.20654788642185527333516964842, 6.81132953531931430246541974082, 7.60760274764492804996292881954, 8.287086326222519113161816348626, 9.116874323884118177732056541440