| L(s) = 1 | + (−3.60 + 1.73i)2-s + (12.1 + 15.2i)3-s + (9.97 − 12.5i)4-s + (−48.0 + 23.1i)5-s + (−70.0 − 33.7i)6-s + (−151. − 190. i)7-s + (−14.2 + 62.3i)8-s + (−30.0 + 131. i)9-s + (133. − 166. i)10-s + (105. + 464. i)11-s + 311.·12-s + (−159. − 697. i)13-s + (876. + 422. i)14-s + (−934. − 450. i)15-s + (−56.9 − 249. i)16-s − 1.27e3·17-s + ⋯ |
| L(s) = 1 | + (−0.637 + 0.306i)2-s + (0.777 + 0.975i)3-s + (0.311 − 0.390i)4-s + (−0.859 + 0.414i)5-s + (−0.794 − 0.382i)6-s + (−1.16 − 1.46i)7-s + (−0.0786 + 0.344i)8-s + (−0.123 + 0.542i)9-s + (0.420 − 0.527i)10-s + (0.264 + 1.15i)11-s + 0.623·12-s + (−0.261 − 1.14i)13-s + (1.19 + 0.575i)14-s + (−1.07 − 0.516i)15-s + (−0.0556 − 0.243i)16-s − 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.106609 - 0.156116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106609 - 0.156116i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.60 - 1.73i)T \) |
| 29 | \( 1 + (2.63e3 + 3.68e3i)T \) |
| good | 3 | \( 1 + (-12.1 - 15.2i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (48.0 - 23.1i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (151. + 190. i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-105. - 464. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (159. + 697. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + 1.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.10e3 + 1.38e3i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (1.22e3 + 592. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (7.43e3 - 3.58e3i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (1.58e3 - 6.94e3i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 - 9.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.76e3 + 847. i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (1.28e3 + 5.64e3i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (3.25e4 - 1.56e4i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + 3.66e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (1.81e4 + 2.28e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-616. + 2.70e3i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-2.90e3 - 1.27e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-1.03e3 - 498. i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (1.29e4 - 5.65e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (4.72e4 - 5.92e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (2.25e4 - 1.08e4i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-1.12e5 + 1.41e5i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
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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
−14.16765805635480926700463211730, −12.82464396269470652498598463626, −11.00460353876027596076991594997, −10.04293656911384626521115726312, −9.334219836128830340485313800135, −7.69641790243123629090696742739, −6.87057089321993719146268698256, −4.33373314258571090828992667195, −3.20068694194071807135267602456, −0.096307767369021440740637939468,
2.01845131538490300673987341576, 3.45167407762936850812259212854, 6.17225206871472045760549614522, 7.55286237899539601851523844872, 8.748309967209235380728917039038, 9.256317155019628607198179575865, 11.39571134728377297286569276656, 12.27279796762647042652396031987, 13.09126897667131944306995267360, 14.42125369089301723279883792819
