| L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.282 − 1.70i)3-s + (−0.951 − 0.309i)4-s + (−0.103 − 2.23i)5-s + (−1.73 + 0.0119i)6-s + (2.28 + 2.28i)7-s + (−0.453 + 0.891i)8-s + (−2.84 + 0.966i)9-s + (−2.22 − 0.247i)10-s + (−2.43 − 3.35i)11-s + (−0.259 + 1.71i)12-s + (−2.10 + 0.333i)13-s + (2.61 − 1.90i)14-s + (−3.78 + 0.808i)15-s + (0.809 + 0.587i)16-s + (4.19 + 2.13i)17-s + ⋯ |
| L(s) = 1 | + (0.110 − 0.698i)2-s + (−0.163 − 0.986i)3-s + (−0.475 − 0.154i)4-s + (−0.0463 − 0.998i)5-s + (−0.707 + 0.00487i)6-s + (0.864 + 0.864i)7-s + (−0.160 + 0.315i)8-s + (−0.946 + 0.322i)9-s + (−0.702 − 0.0781i)10-s + (−0.735 − 1.01i)11-s + (−0.0748 + 0.494i)12-s + (−0.583 + 0.0923i)13-s + (0.699 − 0.508i)14-s + (−0.977 + 0.208i)15-s + (0.202 + 0.146i)16-s + (1.01 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.443731 - 0.975844i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.443731 - 0.975844i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 + 0.987i)T \) |
| 3 | \( 1 + (0.282 + 1.70i)T \) |
| 5 | \( 1 + (0.103 + 2.23i)T \) |
| good | 7 | \( 1 + (-2.28 - 2.28i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.43 + 3.35i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.10 - 0.333i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-4.19 - 2.13i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-7.55 + 2.45i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 0.530i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.0799 + 0.246i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.605 + 1.86i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.925 - 5.84i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.94 - 4.04i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.41 + 6.41i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.85 - 3.63i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (6.16 - 3.14i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (5.82 + 4.23i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (5.87 - 4.27i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.94 - 3.81i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-10.3 - 3.35i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.126 + 0.800i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (3.93 + 1.28i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.20 - 10.2i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (13.0 - 9.48i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.292 + 0.149i)T + (57.0 - 78.4i)T^{2} \) |
| 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
−12.46815544735664087990470163392, −11.84249158497655023996768499550, −11.07172313797857160931891241970, −9.459305186663384248658049406296, −8.418500562750029280116903336979, −7.70383034049984200469936771893, −5.65238452497130904775301736441, −5.09959596808193213476811680929, −2.86230316376272083866335843608, −1.22622763177645282008882261473,
3.21708867025892302664624583118, 4.63147870379684442414235885401, 5.54438067251553651343784139640, 7.28063932653636592493702164704, 7.75025235208069773529379203971, 9.546499089946138684732995497442, 10.23627685792204151560011134533, 11.15943842460738328201106871588, 12.29231994644962715145150418458, 14.01920776152688952040742510206
