| L(s) = 1 | + (1.96 + 1.13i)2-s + (−2.46 + 4.57i)3-s + (−1.41 − 2.45i)4-s + (21.4 + 5.74i)5-s + (−10.0 + 6.19i)6-s + (17.3 − 4.65i)7-s − 24.6i·8-s + (−14.8 − 22.5i)9-s + (35.6 + 35.6i)10-s + (15.2 − 4.08i)11-s + (14.6 − 0.416i)12-s + (20.2 + 35.0i)13-s + (39.4 + 10.5i)14-s + (−79.1 + 83.7i)15-s + (16.6 − 28.8i)16-s + (−63.1 + 30.4i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.401i)2-s + (−0.475 + 0.879i)3-s + (−0.176 − 0.306i)4-s + (1.91 + 0.513i)5-s + (−0.684 + 0.421i)6-s + (0.937 − 0.251i)7-s − 1.08i·8-s + (−0.548 − 0.836i)9-s + (1.12 + 1.12i)10-s + (0.417 − 0.111i)11-s + (0.353 − 0.0100i)12-s + (0.431 + 0.746i)13-s + (0.753 + 0.201i)14-s + (−1.36 + 1.44i)15-s + (0.260 − 0.451i)16-s + (−0.900 + 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.44187 + 1.32451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44187 + 1.32451i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.46 - 4.57i)T \) |
| 17 | \( 1 + (63.1 - 30.4i)T \) |
| good | 2 | \( 1 + (-1.96 - 1.13i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-21.4 - 5.74i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (-17.3 + 4.65i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-15.2 + 4.08i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-20.2 - 35.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 - 54.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (7.18 - 26.8i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (42.1 + 157. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (155. + 41.6i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-82.8 + 82.8i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (37.5 - 139. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (399. + 230. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-80.5 + 139. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 37.6iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-94.3 + 54.4i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-399. + 106. i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (-24.9 - 43.1i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (365. - 365. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-654. + 654. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-35.9 + 9.63i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (1.08e3 + 625. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (146. + 546. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
| 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
−13.10353264657902850609313854950, −11.43897867031810243265598476478, −10.54129975205891553039683313081, −9.750971582554114092166702313478, −8.933791644738991894161664049571, −6.66790697118196101057259612990, −5.96859228688140913313245039549, −5.11287400274927628806543041001, −3.94495768352281351174134553004, −1.70598330145019686268728681849,
1.50329697657337612853863936536, 2.53732041594356284508688062045, 4.91320946007632744170733503639, 5.46657256406596612605995045982, 6.68992213257263790692280831520, 8.310146677053635441426785907319, 9.109299785043565939735002365505, 10.69192379971926676137266236438, 11.55007342239103979639471523880, 12.71814367255031837472938311253
