| L(s) = 1 | + (−0.786 + 0.755i)2-s + (1.73 + 2.31i)3-s + (−0.0326 + 0.810i)4-s + (−1.81 + 1.30i)5-s + (−3.11 − 0.505i)6-s + (4.58 − 0.937i)7-s + (−2.03 − 2.29i)8-s + (−1.49 + 5.15i)9-s + (0.446 − 2.39i)10-s + (1.14 + 3.96i)11-s + (−1.93 + 1.33i)12-s + (−1.10 − 3.43i)13-s + (−2.90 + 4.20i)14-s + (−6.16 − 1.94i)15-s + (1.71 + 0.138i)16-s + (−4.09 + 0.836i)17-s + ⋯ |
| L(s) = 1 | + (−0.556 + 0.534i)2-s + (1.00 + 1.33i)3-s + (−0.0163 + 0.405i)4-s + (−0.813 + 0.582i)5-s + (−1.27 − 0.206i)6-s + (1.73 − 0.354i)7-s + (−0.718 − 0.811i)8-s + (−0.497 + 1.71i)9-s + (0.141 − 0.757i)10-s + (0.346 + 1.19i)11-s + (−0.557 + 0.384i)12-s + (−0.306 − 0.951i)13-s + (−0.775 + 1.12i)14-s + (−1.59 − 0.501i)15-s + (0.428 + 0.0345i)16-s + (−0.993 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.189588 - 1.42275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189588 - 1.42275i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.81 - 1.30i)T \) |
| 13 | \( 1 + (1.10 + 3.43i)T \) |
| good | 2 | \( 1 + (0.786 - 0.755i)T + (0.0805 - 1.99i)T^{2} \) |
| 3 | \( 1 + (-1.73 - 2.31i)T + (-0.834 + 2.88i)T^{2} \) |
| 7 | \( 1 + (-4.58 + 0.937i)T + (6.43 - 2.74i)T^{2} \) |
| 11 | \( 1 + (-1.14 - 3.96i)T + (-9.29 + 5.87i)T^{2} \) |
| 17 | \( 1 + (4.09 - 0.836i)T + (15.6 - 6.66i)T^{2} \) |
| 19 | \( 1 + (2.21 - 3.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.214 - 0.123i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.08 + 2.17i)T + (-1.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-1.35 - 3.56i)T + (-23.2 + 20.5i)T^{2} \) |
| 37 | \( 1 + (-5.84 - 4.77i)T + (7.40 + 36.2i)T^{2} \) |
| 41 | \( 1 + (6.76 - 5.08i)T + (11.4 - 39.3i)T^{2} \) |
| 43 | \( 1 + (2.55 - 2.08i)T + (8.60 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-4.79 + 9.13i)T + (-26.6 - 38.6i)T^{2} \) |
| 53 | \( 1 + (3.05 + 3.45i)T + (-6.38 + 52.6i)T^{2} \) |
| 59 | \( 1 + (-1.26 + 0.102i)T + (58.2 - 9.46i)T^{2} \) |
| 61 | \( 1 + (-11.6 + 3.88i)T + (48.7 - 36.6i)T^{2} \) |
| 67 | \( 1 + (-0.672 + 0.0270i)T + (66.7 - 5.39i)T^{2} \) |
| 71 | \( 1 + (-13.1 - 5.61i)T + (49.1 + 51.2i)T^{2} \) |
| 73 | \( 1 + (-0.933 + 3.78i)T + (-64.6 - 33.9i)T^{2} \) |
| 79 | \( 1 + (12.4 + 6.54i)T + (44.8 + 65.0i)T^{2} \) |
| 83 | \( 1 + (-2.38 - 0.289i)T + (80.5 + 19.8i)T^{2} \) |
| 89 | \( 1 + (-4.85 - 8.41i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 4.93i)T + (61.3 - 75.1i)T^{2} \) |
| 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
−10.33244151588872297297551284634, −9.826566323922903121710996213312, −8.611079545795784861695029698820, −8.201673465886337574201060975964, −7.68789480101968676576003750989, −6.75542867074757656888330129217, −4.92343523235202155575224910923, −4.25916640759447531519621114203, −3.55050629328649809257489507476, −2.26917118780156933303765623798,
0.76602057172815362115300663710, 1.78623321241750122344541399033, 2.54793324604823856134696397322, 4.17147470074266793586333063337, 5.25464779564867538822019725963, 6.50648524241427194196071046457, 7.52209913846873603477184211778, 8.294461430040344829573665620886, 8.859843745222740072314919139077, 9.121979359570790610956180098023
