| L(s) = 1 | + (−21.4 + 7.79i)3-s + (5.95 − 33.7i)5-s + (−60.0 − 103. i)7-s + (211. − 177. i)9-s + (−129. + 223. i)11-s + (228. + 83.3i)13-s + (135. + 769. i)15-s + (723. + 606. i)17-s + (474. + 1.50e3i)19-s + (2.09e3 + 1.75e3i)21-s + (457. + 2.59e3i)23-s + (1.83e3 + 666. i)25-s + (−381. + 661. i)27-s + (1.11e3 − 933. i)29-s + (−2.59e3 − 4.48e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.500i)3-s + (0.106 − 0.603i)5-s + (−0.463 − 0.802i)7-s + (0.871 − 0.731i)9-s + (−0.321 + 0.556i)11-s + (0.375 + 0.136i)13-s + (0.155 + 0.882i)15-s + (0.607 + 0.509i)17-s + (0.301 + 0.953i)19-s + (1.03 + 0.870i)21-s + (0.180 + 1.02i)23-s + (0.586 + 0.213i)25-s + (−0.100 + 0.174i)27-s + (0.245 − 0.206i)29-s + (−0.484 − 0.838i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.818409 + 0.367935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.818409 + 0.367935i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-474. - 1.50e3i)T \) |
| good | 3 | \( 1 + (21.4 - 7.79i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (-5.95 + 33.7i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (60.0 + 103. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (129. - 223. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-228. - 83.3i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-723. - 606. i)T + (2.46e5 + 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-457. - 2.59e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (-1.11e3 + 933. i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (2.59e3 + 4.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 9.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-998. + 363. i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (3.65e3 - 2.07e4i)T + (-1.38e8 - 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-6.16e3 + 5.17e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + (32.9 + 186. i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-1.71e4 - 1.43e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-311. - 1.76e3i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-1.52e4 + 1.28e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-9.23e3 + 5.23e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + (2.73e4 - 9.94e3i)T + (1.58e9 - 1.33e9i)T^{2} \) |
| 79 | \( 1 + (2.00e4 - 7.30e3i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-5.01e4 - 8.68e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-8.06e4 - 2.93e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-7.02e4 - 5.89e4i)T + (1.49e9 + 8.45e9i)T^{2} \) |
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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
−13.39231882280346784796882555359, −12.48736152097970187786290722406, −11.41425775605738709598900659159, −10.37018007334496123191852745367, −9.559987719095081174560527358196, −7.73019417641382187067132303619, −6.25150125975894796430481929042, −5.18655377606942377597283371891, −3.92952949053828907576549461779, −1.01801224355750369785169220230,
0.61456885656173498113490564961, 2.84689438614839780784489630257, 5.18716560422136339081927273354, 6.16625443375461953910589642240, 7.10051386005428480000253011462, 8.828469940931244220551465161475, 10.39297739872143671019775639096, 11.24461012557460777104428176838, 12.20218301116979087488481103942, 13.08327421243101837392557781478
