| L(s) = 1 | + (−9.74 + 16.8i)5-s + (18.4 + 1.67i)7-s + (15.1 − 8.72i)11-s + 16.5i·13-s + (68.6 + 118. i)17-s + (20.6 + 11.9i)19-s + (108. + 62.8i)23-s + (−127. − 220. i)25-s − 105. i·29-s + (95.0 − 54.8i)31-s + (−207. + 294. i)35-s + (58.5 − 101. i)37-s + 348.·41-s + 141.·43-s + (−172. + 299. i)47-s + ⋯ |
| L(s) = 1 | + (−0.871 + 1.50i)5-s + (0.995 + 0.0902i)7-s + (0.414 − 0.239i)11-s + 0.352i·13-s + (0.979 + 1.69i)17-s + (0.249 + 0.144i)19-s + (0.987 + 0.570i)23-s + (−1.01 − 1.76i)25-s − 0.673i·29-s + (0.550 − 0.318i)31-s + (−1.00 + 1.42i)35-s + (0.260 − 0.450i)37-s + 1.32·41-s + 0.503·43-s + (−0.536 + 0.929i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.078639331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078639331\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-18.4 - 1.67i)T \) |
| good | 5 | \( 1 + (9.74 - 16.8i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-15.1 + 8.72i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 16.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-68.6 - 118. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-20.6 - 11.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-108. - 62.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 105. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-95.0 + 54.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-58.5 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (172. - 299. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-149. + 86.1i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-297. - 515. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (561. + 324. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-64.9 - 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 908. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (44.8 - 25.9i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (474. - 822. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (210. - 364. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 553. iT - 9.12e5T^{2} \) |
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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
−10.05360669603224092783840118543, −8.910936099509994473114500090198, −7.84563484923947399966544069777, −7.59590884495377197523980730763, −6.47643531150801816757110774011, −5.72728084803340618839450543834, −4.31095042865317815512692425560, −3.63066234095618383364351506274, −2.57839358916078353764797216014, −1.24356314283796044322730426051,
0.63578945973081840651196700084, 1.28964411214958552052947750483, 2.96401995201071206609107053114, 4.26339411748107880878088903083, 4.89109905215752336195503037300, 5.48926032767211654623330462751, 7.12167976308807408653217264408, 7.70195823345726235736783573763, 8.552102591786979265009238989567, 9.089526707505043863304790686919
