| L(s) = 1 | + (−0.812 + 0.469i)5-s + (1.05 − 6.92i)7-s + (−13.4 − 7.73i)11-s − 9.93·13-s + (−8.79 − 5.07i)17-s + (17.6 + 30.6i)19-s + (8.79 − 5.07i)23-s + (−12.0 + 20.8i)25-s + 51.2i·29-s + (14.4 − 25.0i)31-s + (2.39 + 6.11i)35-s + (8.41 + 14.5i)37-s + 38.6i·41-s + 66.3·43-s + (−36.4 + 21.0i)47-s + ⋯ |
| L(s) = 1 | + (−0.162 + 0.0938i)5-s + (0.150 − 0.988i)7-s + (−1.21 − 0.703i)11-s − 0.764·13-s + (−0.517 − 0.298i)17-s + (0.931 + 1.61i)19-s + (0.382 − 0.220i)23-s + (−0.482 + 0.835i)25-s + 1.76i·29-s + (0.465 − 0.806i)31-s + (0.0683 + 0.174i)35-s + (0.227 + 0.393i)37-s + 0.943i·41-s + 1.54·43-s + (−0.774 + 0.447i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7578656922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7578656922\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.05 + 6.92i)T \) |
| good | 5 | \( 1 + (0.812 - 0.469i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (13.4 + 7.73i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 9.93T + 169T^{2} \) |
| 17 | \( 1 + (8.79 + 5.07i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-17.6 - 30.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-8.79 + 5.07i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 51.2iT - 841T^{2} \) |
| 31 | \( 1 + (-14.4 + 25.0i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-8.41 - 14.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 38.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 66.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (36.4 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (0.210 + 0.121i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (53.1 + 30.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-16.6 - 28.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-13.4 + 23.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 29.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (11.0 - 19.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-35.6 - 61.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (148. - 85.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 83.3T + 9.40e3T^{2} \) |
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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
−10.06230480311692079015767735008, −9.317395464376301530983193986227, −7.992874859668245393631416605613, −7.71646135066101415730140300180, −6.75206837176441007601727490507, −5.59639906275567377631429691256, −4.82526589141239539371526143049, −3.68735956109079631138102501170, −2.76495839523969253333346632272, −1.20551368407394786204791651269,
0.24799020558375639357065010610, 2.21300680340790048000968307598, 2.80778930660616766484768788013, 4.46342334367851361621644963644, 5.09015227021216579100060202644, 6.00287769866100468383501285810, 7.16796182653575240457153973417, 7.81697326040274104142788725174, 8.751076525504740881982353249660, 9.513603966953476519113872065291
