| L(s) = 1 | + (0.415 − 0.909i)2-s + (−2.49 − 0.733i)3-s + (−0.654 − 0.755i)4-s + (−1.90 + 1.22i)5-s + (−1.70 + 1.96i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (3.17 + 2.03i)9-s + (0.322 + 2.24i)10-s + (2.57 + 5.63i)11-s + (1.08 + 2.36i)12-s + (−0.405 − 2.82i)13-s + (−0.841 − 0.540i)14-s + (5.65 − 1.66i)15-s + (−0.142 + 0.989i)16-s + (−3.20 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−1.44 − 0.423i)3-s + (−0.327 − 0.377i)4-s + (−0.852 + 0.547i)5-s + (−0.695 + 0.802i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (1.05 + 0.679i)9-s + (0.101 + 0.709i)10-s + (0.775 + 1.69i)11-s + (0.312 + 0.683i)12-s + (−0.112 − 0.782i)13-s + (−0.224 − 0.144i)14-s + (1.46 − 0.428i)15-s + (−0.0355 + 0.247i)16-s + (−0.776 + 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436871 + 0.211018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436871 + 0.211018i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-3.07 + 3.68i)T \) |
| good | 3 | \( 1 + (2.49 + 0.733i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (1.90 - 1.22i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-2.57 - 5.63i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.405 + 2.82i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (3.20 - 3.69i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.80 - 4.38i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (4.57 - 5.28i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.66 - 1.95i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (7.09 + 4.56i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.48 - 0.953i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-7.43 - 2.18i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 0.254T + 47T^{2} \) |
| 53 | \( 1 + (1.47 - 10.2i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.441 + 3.07i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (6.24 - 1.83i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.96 - 10.8i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.394 + 0.863i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.31 + 2.67i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.28 + 8.96i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.47 - 3.52i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 3.72i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (2.77 - 1.78i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.87863284060334680540551489383, −10.74745245171973314684089450548, −10.61490534322597011368203007959, −9.215419525277135090456499010468, −7.49899805705963236053566092276, −6.96399769122544177620149178023, −5.76063651738149309856232325072, −4.65522838324436220467180836728, −3.62579133842864361630440300624, −1.56477189037074097752269729662,
0.39826475703716875376895864652, 3.60645545115095803375649185161, 4.69997338951276209426164731310, 5.47864752211213254254702788471, 6.41244037874701302295485903831, 7.40290433693687716253651647035, 8.781565855272976731638606355020, 9.358379208312109506710299452283, 11.14065014340670960587039323930, 11.49097074480481663870063484272
