| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (0.705 − 1.22i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (2.76 + 4.79i)11-s + (−0.499 + 0.866i)12-s + (−4.69 + 1.71i)13-s + (0.245 + 1.39i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (5.94 − 4.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (0.383 − 0.139i)6-s + (0.266 − 0.462i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (0.833 + 1.44i)11-s + (−0.144 + 0.250i)12-s + (−1.30 + 0.474i)13-s + (0.0655 + 0.371i)14-s + (−0.0448 + 0.254i)15-s + (−0.234 − 0.0855i)16-s + (1.44 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.833052 - 0.303514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833052 - 0.303514i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (2.23 + 3.74i)T \) |
| good | 7 | \( 1 + (-0.705 + 1.22i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.76 - 4.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.69 - 1.71i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.94 + 4.98i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.44 + 8.20i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.34 - 1.13i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.71 + 4.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + (-4.89 - 1.78i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.837 + 4.74i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.04 - 1.71i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.52 + 8.66i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (5.24 - 4.39i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.67 + 9.49i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.29 + 4.43i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.361 + 2.05i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.800 + 0.291i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (9.06 + 3.29i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.18 - 2.05i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.3 + 4.86i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 9.31i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.41304681910638229006280878318, −9.711619782581365598209843985478, −9.023942876544435882567517625244, −7.65944383323075917910678140619, −7.20449953901817950778736849845, −6.33558874803944169939883522172, −4.85352919546511728205953365283, −4.54695525907449977564774229488, −2.29449859182110137903216335871, −0.75013417158504110099771127527,
1.30155696186020984279144278043, 2.99528263254116291022937265793, 3.92776683667664695126528704876, 5.49493137974040945958901498448, 6.14120762502998551169434007876, 7.47138661293630882479568128030, 8.219193473987804738330140544951, 9.225985267337741142389947675665, 10.15139336178511050471966040558, 10.71010753085958839280257964083
