| L(s) = 1 | + (−3.99 − 0.0325i)2-s + (−1.55 + 0.273i)3-s + (15.9 + 0.260i)4-s + (3.23 − 2.71i)5-s + (6.21 − 1.04i)6-s + (4.97 + 2.87i)7-s + (−63.9 − 1.56i)8-s + (−73.7 + 26.8i)9-s + (−13.0 + 10.7i)10-s + (133. − 77.0i)11-s + (−24.8 + 3.97i)12-s + (−4.87 + 27.6i)13-s + (−19.7 − 11.6i)14-s + (−4.27 + 5.09i)15-s + (255. + 8.32i)16-s + (485. + 176. i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.00813i)2-s + (−0.172 + 0.0304i)3-s + (0.999 + 0.0162i)4-s + (0.129 − 0.108i)5-s + (0.172 − 0.0289i)6-s + (0.101 + 0.0585i)7-s + (−0.999 − 0.0243i)8-s + (−0.910 + 0.331i)9-s + (−0.130 + 0.107i)10-s + (1.10 − 0.636i)11-s + (−0.172 + 0.0275i)12-s + (−0.0288 + 0.163i)13-s + (−0.100 − 0.0594i)14-s + (−0.0189 + 0.0226i)15-s + (0.999 + 0.0325i)16-s + (1.68 + 0.611i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.05594 - 0.0657323i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05594 - 0.0657323i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.99 + 0.0325i)T \) |
| 19 | \( 1 + (-250. + 259. i)T \) |
| good | 3 | \( 1 + (1.55 - 0.273i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-3.23 + 2.71i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-4.97 - 2.87i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-133. + 77.0i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (4.87 - 27.6i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-485. - 176. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-509. + 607. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-722. + 262. i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-881. - 508. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.27e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (91.5 + 518. i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (658. + 784. i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-855. - 2.34e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.56e3 - 1.31e3i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-938. + 2.57e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-719. - 603. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-2.40e3 - 6.61e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (5.80e3 + 6.92e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (76.9 + 436. i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (5.31e3 - 936. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-5.68e3 - 3.27e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.16e3 - 6.62e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (8.55e3 + 3.11e3i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−13.95680480565224066627686782645, −12.20876252132081520301904125664, −11.43143585032752072759516228857, −10.35588764235288779681646636076, −9.081757867901396933616056667757, −8.240230776443435598951615425608, −6.76078489988882177069055243603, −5.51450909848281843385503583570, −3.08954838014528141092005472868, −1.04504918605078040398811351316,
1.12936070732712665604077882224, 3.16794088873074339220918568249, 5.60104611601969541898560132578, 6.87769909715079070428527934469, 8.093672668384618436559111381634, 9.362991986544818577206342090230, 10.19262637909386348211877089992, 11.71216497374226016744291841437, 12.07724216828185554063072847063, 14.09496666133414195464819855205
