L(s) = 1 | + (0.182 + 0.315i)3-s + (0.5 + 0.866i)5-s + 0.635·7-s + (1.43 − 2.48i)9-s + 1.63·11-s + (0.5 − 0.866i)13-s + (−0.182 + 0.315i)15-s + (3.29 + 5.71i)17-s + (0.0466 + 4.35i)19-s + (0.115 + 0.200i)21-s + (−0.433 + 0.750i)23-s + (−0.499 + 0.866i)25-s + 2.13·27-s + (4.54 − 7.87i)29-s − 1.86·31-s + ⋯ |
L(s) = 1 | + (0.105 + 0.182i)3-s + (0.223 + 0.387i)5-s + 0.240·7-s + (0.477 − 0.827i)9-s + 0.493·11-s + (0.138 − 0.240i)13-s + (−0.0470 + 0.0814i)15-s + (0.799 + 1.38i)17-s + (0.0107 + 0.999i)19-s + (0.0252 + 0.0437i)21-s + (−0.0904 + 0.156i)23-s + (−0.0999 + 0.173i)25-s + 0.411·27-s + (0.844 − 1.46i)29-s − 0.335·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56135 + 0.253133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56135 + 0.253133i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.0466 - 4.35i)T \) |
good | 3 | \( 1 + (-0.182 - 0.315i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 0.635T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.29 - 5.71i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (0.433 - 0.750i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.54 + 7.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 - 0.635T + 37T^{2} \) |
| 41 | \( 1 + (0.953 + 1.65i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.98 + 3.42i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.54 - 7.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.93 + 8.54i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.54 + 7.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.23 - 10.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.201 - 0.349i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.36 + 9.29i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + (0.271 - 0.469i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.68 - 6.38i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.43575141098723532179203858962, −10.23177288492520087847564532532, −9.837400515738180329777063162952, −8.608918951875006508990790692233, −7.74352799615561963314000426820, −6.50338288743189531476504583240, −5.79809858957910554948115090876, −4.23063384828580331039131239417, −3.34645317927965340497117211025, −1.56030245347418854257566038805,
1.39284869385322592627459890052, 2.90526277020918222790906670142, 4.55055626786396321198400741879, 5.27365491749742548685100334404, 6.71608937892576122928682026641, 7.51257092743442069735255700395, 8.591811352899394324987458267443, 9.420058305943828674969398132313, 10.37882561173395505709602636551, 11.36274003782228124123097282992