| L(s) = 1 | + (3.75 + 1.36i)2-s + (−2.69 − 15.2i)3-s + (12.2 + 10.2i)4-s + (−6.46 + 5.42i)5-s + (10.7 − 61.0i)6-s + (110. − 191. i)7-s + (32.0 + 55.4i)8-s + (2.64 − 0.963i)9-s + (−31.7 + 11.5i)10-s + (−362. − 628. i)11-s + (123. − 214. i)12-s + (−87.6 + 497. i)13-s + (677. − 568. i)14-s + (100. + 84.0i)15-s + (44.4 + 252. i)16-s + (2.02e3 + 737. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.172 − 0.979i)3-s + (0.383 + 0.321i)4-s + (−0.115 + 0.0969i)5-s + (0.122 − 0.692i)6-s + (0.852 − 1.47i)7-s + (0.176 + 0.306i)8-s + (0.0108 − 0.00396i)9-s + (−0.100 + 0.0364i)10-s + (−0.904 − 1.56i)11-s + (0.248 − 0.430i)12-s + (−0.143 + 0.816i)13-s + (0.923 − 0.774i)14-s + (0.114 + 0.0964i)15-s + (0.0434 + 0.246i)16-s + (1.70 + 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.90436 - 1.15483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90436 - 1.15483i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.75 - 1.36i)T \) |
| 19 | \( 1 + (1.01e3 - 1.20e3i)T \) |
| good | 3 | \( 1 + (2.69 + 15.2i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (6.46 - 5.42i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-110. + 191. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (362. + 628. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (87.6 - 497. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-2.02e3 - 737. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-1.47e3 - 1.23e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (2.79e3 - 1.01e3i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (636. - 1.10e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 8.22e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (373. + 2.11e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-755. + 634. i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-9.60e3 + 3.49e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-4.91e3 - 4.12e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-2.60e4 - 9.49e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-5.05e3 - 4.24e3i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (5.65e4 - 2.05e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-8.35e3 + 7.01e3i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-1.76e3 - 1.00e4i)T + (-1.94e9 + 7.09e8i)T^{2} \) |
| 79 | \( 1 + (-1.70e4 - 9.67e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-3.80e4 + 6.58e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.16e4 + 6.59e4i)T + (-5.24e9 - 1.90e9i)T^{2} \) |
| 97 | \( 1 + (7.95e3 + 2.89e3i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−14.69768557448494891733064615424, −13.77349692912369076456415133222, −12.98367157103286528463409579875, −11.58920896267019058888639967971, −10.54787841296901390527793311151, −8.009674119752539308002333878136, −7.26001890515885028171017906961, −5.69740578364216654379244386410, −3.78538906374365782021042134776, −1.24137473044716588324291455402,
2.48468222736293128356696143534, 4.67661426224759136456737794185, 5.41318460674539073246427406773, 7.76982881659937665453934405163, 9.551757519819219640705631555260, 10.62112388129887731730558818370, 11.98877610784813942963589195718, 12.83875241445129036011433453246, 14.92234331518908983381398206269, 15.08406884311718242994686357675
