L(s) = 1 | + (−4.66 + 2.29i)3-s + (−5.16 + 8.94i)5-s + (14.4 − 11.5i)7-s + (16.4 − 21.3i)9-s + (27.1 − 15.6i)11-s + (39.0 + 22.5i)13-s + (3.57 − 53.5i)15-s − 62.4·17-s + 132. i·19-s + (−41.1 + 86.9i)21-s + (−58.8 − 33.9i)23-s + (9.14 + 15.8i)25-s + (−27.8 + 137. i)27-s + (−116. + 67.3i)29-s + (25.9 + 15.0i)31-s + ⋯ |
L(s) = 1 | + (−0.897 + 0.441i)3-s + (−0.461 + 0.800i)5-s + (0.782 − 0.622i)7-s + (0.610 − 0.791i)9-s + (0.743 − 0.429i)11-s + (0.832 + 0.480i)13-s + (0.0616 − 0.921i)15-s − 0.891·17-s + 1.60i·19-s + (−0.428 + 0.903i)21-s + (−0.533 − 0.307i)23-s + (0.0731 + 0.126i)25-s + (−0.198 + 0.980i)27-s + (−0.747 + 0.431i)29-s + (0.150 + 0.0869i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.139631167\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139631167\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.66 - 2.29i)T \) |
| 7 | \( 1 + (-14.4 + 11.5i)T \) |
good | 5 | \( 1 + (5.16 - 8.94i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-27.1 + 15.6i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-39.0 - 22.5i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 132. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (58.8 + 33.9i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (116. - 67.3i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-25.9 - 15.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 40.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + (39.7 - 68.8i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-161. - 279. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-171. - 296. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 64.9iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (79.3 - 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (493. - 285. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-150. + 261. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 719. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 558. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-456. - 790. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-352. - 610. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 700.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-202. + 117. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.44346073318422401978776657284, −11.10373922621359914914778302352, −10.30005177736889008327454412318, −9.054476308935793905034230508065, −7.81457656616289663719158168584, −6.72451194305531149288544940678, −5.90155160618577079927541080266, −4.36641670794985176137599933526, −3.66272804635094171745634110463, −1.34689005690750957041447709361,
0.57268994883659572784482335795, 1.97397163792564986236529579602, 4.24431089783476898768839033693, 5.09283338015685056043931602662, 6.18408616493005608313814676487, 7.32719063342222864898675120635, 8.416062134330626369505768227903, 9.205160534401851650094253882299, 10.74695166430119442732479351221, 11.49456947411706834630514967018