| L(s) = 1 | + (−0.453 − 0.891i)2-s + (−1.06 + 1.36i)3-s + (−0.587 + 0.809i)4-s + (0.483 − 2.18i)5-s + (1.70 + 0.323i)6-s + (3.13 + 3.13i)7-s + (0.987 + 0.156i)8-s + (−0.750 − 2.90i)9-s + (−2.16 + 0.560i)10-s + (3.57 + 1.16i)11-s + (−0.484 − 1.66i)12-s + (3.49 + 1.78i)13-s + (1.37 − 4.21i)14-s + (2.47 + 2.97i)15-s + (−0.309 − 0.951i)16-s + (−0.0644 + 0.406i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (−0.612 + 0.790i)3-s + (−0.293 + 0.404i)4-s + (0.216 − 0.976i)5-s + (0.694 + 0.132i)6-s + (1.18 + 1.18i)7-s + (0.349 + 0.0553i)8-s + (−0.250 − 0.968i)9-s + (−0.684 + 0.177i)10-s + (1.07 + 0.350i)11-s + (−0.139 − 0.480i)12-s + (0.969 + 0.494i)13-s + (0.366 − 1.12i)14-s + (0.639 + 0.768i)15-s + (−0.0772 − 0.237i)16-s + (−0.0156 + 0.0986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.916312 - 0.0267000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.916312 - 0.0267000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 3 | \( 1 + (1.06 - 1.36i)T \) |
| 5 | \( 1 + (-0.483 + 2.18i)T \) |
| good | 7 | \( 1 + (-3.13 - 3.13i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.57 - 1.16i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.49 - 1.78i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.0644 - 0.406i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (2.93 + 4.04i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.48 - 2.28i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (1.49 + 1.08i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.69 + 1.96i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.39 - 4.70i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (3.24 - 1.05i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.71 + 3.71i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.10 - 0.808i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.29 + 8.20i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 5.34i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.43 + 13.6i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.11 + 0.968i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-0.992 + 1.36i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.62 + 3.19i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (6.24 - 8.60i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (7.00 + 1.10i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (0.324 - 0.999i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.20 + 7.61i)T + (-92.2 + 29.9i)T^{2} \) |
| 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
−12.59040614528214043821229497361, −11.64215942110058897630857712076, −11.37263765663171503942375099210, −9.859054007742978492913320822342, −8.952709361373339749957866251269, −8.422063164574462204742061668297, −6.21876538072175930973367039011, −5.01493279208235767827544418263, −4.08631150430955698133450987733, −1.71335109634273472226206399731,
1.46143568748132494693905619147, 4.08613072861487925433833840576, 5.80902945640279451505910356702, 6.64914483408092365306237699589, 7.61060758338976749224097099838, 8.451857894980551982760706433188, 10.33397255507174633895032616412, 10.87844105160871528235306692808, 11.80851746350134808538714868718, 13.33380889079216786141233466709
