| L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−1.27 − 2.78i)5-s + (−0.142 + 0.989i)6-s + (0.959 − 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (2.93 + 0.861i)10-s + (2.20 + 2.54i)11-s + (−0.654 − 0.755i)12-s + (5.26 + 1.54i)13-s + (−0.415 + 0.909i)14-s + (−2.57 − 1.65i)15-s + (−0.959 + 0.281i)16-s + (−1.04 + 7.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.568 − 1.24i)5-s + (−0.0580 + 0.404i)6-s + (0.362 − 0.106i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (0.928 + 0.272i)10-s + (0.665 + 0.768i)11-s + (−0.189 − 0.218i)12-s + (1.45 + 0.428i)13-s + (−0.111 + 0.243i)14-s + (−0.664 − 0.427i)15-s + (−0.239 + 0.0704i)16-s + (−0.252 + 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38109 - 0.534879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38109 - 0.534879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-3.99 - 2.64i)T \) |
| good | 5 | \( 1 + (1.27 + 2.78i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 2.54i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.26 - 1.54i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.04 - 7.24i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (1.22 + 8.54i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.37 + 9.55i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (2.84 + 1.83i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.17 + 4.76i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.46 + 5.40i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.71 + 4.96i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 + (-2.04 + 0.599i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.18 + 1.81i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.00 + 1.92i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.83 - 4.42i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (8.11 - 9.36i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.473 + 3.29i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-6.34 - 1.86i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.02 - 6.62i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (12.7 - 8.20i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.52 - 5.52i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.433672102609475212093940319436, −8.852825252285545837166056608544, −8.448761368652825844805533386140, −7.54522922337743535161433341042, −6.69494396090049823939915434180, −5.74025393488629571078642760843, −4.43026260310070504217953408173, −3.99123714488634332105387431258, −1.95930487307774388061479978237, −0.905096500470405977881427811328,
1.36715536993406681481044891763, 3.13600044340887118443859665781, 3.25621817752625073706535655809, 4.49749801419688650641093080045, 5.96360061645007251420804664475, 6.92725297560562191745045358945, 7.78901649880286703848218388151, 8.569429562658639892272305634553, 9.197581017307478959064655675399, 10.33576151662691866334367398911
