| L(s) = 1 | + (−0.342 + 1.49i)2-s + (9.55 − 11.9i)3-s + (26.6 + 12.8i)4-s + (41.6 − 52.2i)5-s + (14.7 + 18.4i)6-s + (−96.6 − 86.3i)7-s + (−59.1 + 74.1i)8-s + (1.80 + 7.90i)9-s + (64.0 + 80.3i)10-s + (14.4 − 63.1i)11-s + (409. − 197. i)12-s + (133. − 584. i)13-s + (162. − 115. i)14-s + (−227. − 998. i)15-s + (500. + 627. i)16-s + (296. − 142. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0605 + 0.265i)2-s + (0.613 − 0.768i)3-s + (0.834 + 0.401i)4-s + (0.745 − 0.934i)5-s + (0.166 + 0.209i)6-s + (−0.745 − 0.666i)7-s + (−0.326 + 0.409i)8-s + (0.00742 + 0.0325i)9-s + (0.202 + 0.254i)10-s + (0.0359 − 0.157i)11-s + (0.820 − 0.395i)12-s + (0.218 − 0.959i)13-s + (0.221 − 0.157i)14-s + (−0.261 − 1.14i)15-s + (0.488 + 0.612i)16-s + (0.248 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.23918 - 0.885633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23918 - 0.885633i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (96.6 + 86.3i)T \) |
| good | 2 | \( 1 + (0.342 - 1.49i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-9.55 + 11.9i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-41.6 + 52.2i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (-14.4 + 63.1i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-133. + 584. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-296. + 142. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 - 949.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-986. - 474. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (6.59e3 - 3.17e3i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + 2.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (2.07e3 - 1.00e3i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (-498. + 625. i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-1.00e4 - 1.26e4i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (2.53e3 - 1.11e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (9.37e3 + 4.51e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (1.91e4 + 2.39e4i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (2.34e4 - 1.12e4i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 - 4.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.82e4 - 1.84e4i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-1.67e4 - 7.35e4i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 + 8.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.87e4 + 8.21e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (8.50e3 + 3.72e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 - 3.97e4T + 8.58e9T^{2} \) |
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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
−14.26228245916516594885631543470, −13.04876895354405699307920434300, −12.72235965583554012153365824202, −10.92596429287617989234636602476, −9.411885689218756609099671155464, −8.053210295418799061708419680755, −7.09408220747796388623825031818, −5.62694221766316386454756660620, −3.09304452073418972552805769368, −1.38452030599462812913412133518,
2.22815970062885146034964768125, 3.43668949508549328217756446902, 5.90172916892797143330297900431, 6.95095521620725954527484902067, 9.233089556095687278146325569183, 9.852847698068428282238643017466, 10.96044406362593494992843628801, 12.25312287477591412251741022629, 13.91626857886626437076180309712, 14.88906245967231805002094287116
