| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−2.78 − 8.55i)3-s + (4.94 + 15.2i)4-s + (−24.3 − 50.3i)5-s + (−11.1 + 34.2i)6-s + 94.8·7-s + (19.7 − 60.8i)8-s + (−65.5 + 47.6i)9-s + (−39.3 + 220. i)10-s + (464. + 337. i)11-s + (116. − 84.6i)12-s + (710. − 516. i)13-s + (−306. − 223. i)14-s + (−362. + 348. i)15-s + (−207. + 150. i)16-s + (−454. + 1.39e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.436 − 0.899i)5-s + (−0.126 + 0.388i)6-s + 0.731·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (−0.124 + 0.696i)10-s + (1.15 + 0.840i)11-s + (0.233 − 0.169i)12-s + (1.16 − 0.847i)13-s + (−0.418 − 0.304i)14-s + (−0.416 + 0.400i)15-s + (−0.202 + 0.146i)16-s + (−0.381 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.492866838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492866838\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 3 | \( 1 + (2.78 + 8.55i)T \) |
| 5 | \( 1 + (24.3 + 50.3i)T \) |
| good | 7 | \( 1 - 94.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-464. - 337. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-710. + 516. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (454. - 1.39e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-607. + 1.87e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-3.84e3 - 2.79e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (19.3 + 59.4i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.48e3 + 7.64e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.00e3 + 4.36e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.94e3 - 5.77e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 8.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.32e3 + 1.02e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.00e4 + 3.09e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-4.05e4 + 2.94e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.50e4 + 1.82e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-3.50e3 + 1.07e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.32e4 - 4.06e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.74e4 - 3.45e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.16e4 + 3.57e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.31e4 - 7.13e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (3.87e4 + 2.81e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.06e4 - 3.26e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.46394779533372190324708800246, −11.30240584419087989622313804589, −9.610277113955357439682263777846, −8.623669020504553202818242043179, −7.88008394300132679453318242567, −6.64614597415447413451889990833, −5.07060810860173741265120399563, −3.73433328963244611856047779188, −1.69671499495121910656906333281, −0.813660914653209056341581175841,
1.15932809153878473972794190460, 3.24393051873161376340180132832, 4.58737358593904684599973165175, 6.16922936609359525288625627102, 6.97870763612752466263586859136, 8.391229606800168409527792000961, 9.115009816831290668503224383080, 10.47104660310923475839190898692, 11.26813660016393539173141629630, 11.79944095453520854152239658323
