| L(s) = 1 | + (3.23 + 2.35i)2-s + (−4.00 − 12.3i)3-s + (4.94 + 15.2i)4-s + (32.6 + 45.3i)5-s + (16.0 − 49.3i)6-s + 24.1·7-s + (−19.7 + 60.8i)8-s + (60.3 − 43.8i)9-s + (−0.889 + 223. i)10-s + (478. + 347. i)11-s + (167. − 122. i)12-s + (715. − 520. i)13-s + (78.3 + 56.8i)14-s + (428. − 585. i)15-s + (−207. + 150. i)16-s + (−489. + 1.50e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.257 − 0.791i)3-s + (0.154 + 0.475i)4-s + (0.584 + 0.811i)5-s + (0.181 − 0.559i)6-s + 0.186·7-s + (−0.109 + 0.336i)8-s + (0.248 − 0.180i)9-s + (−0.00281 + 0.707i)10-s + (1.19 + 0.866i)11-s + (0.336 − 0.244i)12-s + (1.17 − 0.853i)13-s + (0.106 + 0.0775i)14-s + (0.491 − 0.671i)15-s + (−0.202 + 0.146i)16-s + (−0.410 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.39466 + 0.604697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39466 + 0.604697i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 5 | \( 1 + (-32.6 - 45.3i)T \) |
| good | 3 | \( 1 + (4.00 + 12.3i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 - 24.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-478. - 347. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-715. + 520. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (489. - 1.50e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (42.6 - 131. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.64e3 + 1.91e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-614. - 1.89e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.97e3 + 9.14e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (139. - 101. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.06e4 - 7.73e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 118.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.35e3 + 7.24e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (5.17e3 + 1.59e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (8.64e3 - 6.28e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.84e4 - 1.34e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-2.12e4 + 6.53e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.78e4 - 5.50e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.26e3 - 3.82e3i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-4.48e3 - 1.38e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (97.8 - 301. i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (3.97e4 + 2.89e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (3.35e4 + 1.03e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62876689423461251995156249397, −13.45054609557318364713680124214, −12.59730160637453050827567332756, −11.38499438683832177449717739806, −9.976112751293285999125415340756, −8.138841128110561282655187839470, −6.67039564660273343409737646185, −6.10733605746776677340123027538, −3.91217687809439517928454344550, −1.76037844880166632279124758887,
1.42784606663124978420593264335, 3.87297502087691043310202733091, 5.03152556121252546185661591715, 6.38736179255655134426991099208, 8.787649128295562566401325012527, 9.730108419041751189008199583018, 11.10011042437064330369879070510, 11.97182835869749250995841702643, 13.58697530429981537963623172052, 14.04254194156753243605197209807
