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
−15.08406884311718242994686357675, −14.92234331518908983381398206269, −12.83875241445129036011433453246, −11.98877610784813942963589195718, −10.62112388129887731730558818370, −9.551757519819219640705631555260, −7.76982881659937665453934405163, −5.41318460674539073246427406773, −4.67661426224759136456737794185, −2.48468222736293128356696143534,
1.24137473044716588324291455402, 3.78538906374365782021042134776, 5.69740578364216654379244386410, 7.26001890515885028171017906961, 8.009674119752539308002333878136, 10.54787841296901390527793311151, 11.58920896267019058888639967971, 12.98367157103286528463409579875, 13.77349692912369076456415133222, 14.69768557448494891733064615424