| L(s) = 1 | + (1.87 − 0.684i)2-s + (2.73 + 0.996i)3-s + (3.06 − 2.57i)4-s + (3.22 + 18.3i)5-s + 5.82·6-s + (2.30 + 13.0i)7-s + (4.00 − 6.92i)8-s + (−14.1 − 11.8i)9-s + (18.5 + 32.2i)10-s + (18.0 − 31.2i)11-s + (10.9 − 3.98i)12-s + (31.4 − 26.4i)13-s + (13.2 + 23.0i)14-s + (−9.41 + 53.3i)15-s + (2.77 − 15.7i)16-s + (−49.2 − 41.3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.527 + 0.191i)3-s + (0.383 − 0.321i)4-s + (0.288 + 1.63i)5-s + 0.396·6-s + (0.124 + 0.706i)7-s + (0.176 − 0.306i)8-s + (−0.525 − 0.440i)9-s + (0.588 + 1.01i)10-s + (0.494 − 0.856i)11-s + (0.263 − 0.0959i)12-s + (0.671 − 0.563i)13-s + (0.253 + 0.439i)14-s + (−0.161 + 0.918i)15-s + (0.0434 − 0.246i)16-s + (−0.703 − 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.44650 + 0.531173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44650 + 0.531173i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 + 0.684i)T \) |
| 37 | \( 1 + (85.0 - 208. i)T \) |
| good | 3 | \( 1 + (-2.73 - 0.996i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-3.22 - 18.3i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-2.30 - 13.0i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-18.0 + 31.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-31.4 + 26.4i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (49.2 + 41.3i)T + (853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (15.2 + 5.56i)T + (5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-52.3 - 90.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-128. + 221. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 111.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (78.9 - 66.2i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (24.2 + 42.0i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (67.3 - 381. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-54.5 + 309. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-454. + 381. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-165. - 936. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (613. + 223. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 - 962.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-200. - 1.13e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (674. + 565. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-84.0 + 476. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (810. + 1.40e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.11401526606316381340630049151, −13.47895466686979550481564191829, −11.67718610042480289215518102703, −11.10964837754743341392110497323, −9.788586891222191756307846267478, −8.462163992884352545946230437642, −6.69959001601180411201026702336, −5.78174062851161240318751301614, −3.51556242299771364544181675484, −2.63038884760902630606136665917,
1.71684396218126337594150956042, 4.09609630261141830269807182304, 5.16199406069028225482212461248, 6.82300462808547688578958737738, 8.337350765919686201101683859504, 9.058647874839346280202647551923, 10.80400035437808292925459166232, 12.25408040786301011041171315218, 13.07651877070698880295115227321, 13.84835158692650832121862297824
