| L(s) = 1 | + (2.17 − 1.58i)2-s + (1.29 − 0.577i)3-s + (1.61 − 4.98i)4-s + (−0.304 − 0.526i)5-s + (1.90 − 3.30i)6-s + (−1.15 − 1.28i)7-s + (−2.69 − 8.28i)8-s + (−0.659 + 0.732i)9-s + (−1.49 − 0.665i)10-s + (1.30 − 0.278i)11-s + (−0.776 − 7.39i)12-s + (−0.383 + 3.65i)13-s + (−4.54 − 0.965i)14-s + (−0.698 − 0.507i)15-s + (−10.4 − 7.61i)16-s + (2.70 + 0.574i)17-s + ⋯ |
| L(s) = 1 | + (1.53 − 1.11i)2-s + (0.748 − 0.333i)3-s + (0.809 − 2.49i)4-s + (−0.136 − 0.235i)5-s + (0.779 − 1.34i)6-s + (−0.436 − 0.485i)7-s + (−0.951 − 2.92i)8-s + (−0.219 + 0.244i)9-s + (−0.472 − 0.210i)10-s + (0.394 − 0.0838i)11-s + (−0.224 − 2.13i)12-s + (−0.106 + 1.01i)13-s + (−1.21 − 0.258i)14-s + (−0.180 − 0.131i)15-s + (−2.61 − 1.90i)16-s + (0.655 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30341 - 4.13092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30341 - 4.13092i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (-2.17 + 1.58i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.29 + 0.577i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (0.304 + 0.526i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.15 + 1.28i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 0.278i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.383 - 3.65i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-2.70 - 0.574i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.268 - 2.55i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.166 + 0.512i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.57 + 4.77i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0950 + 0.0423i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.314 - 2.98i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-5.44 - 3.95i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.87 + 2.07i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.425 + 0.189i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.18 + 3.53i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (7.34 - 1.56i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-9.48 - 2.01i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (15.2 + 6.78i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (4.73 - 14.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.395 - 1.21i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.01713468783459742050630153532, −9.086209699710909544482442514940, −8.016005593088665840479520743971, −6.80405354256185679919653353392, −6.10089294392030103914016640062, −4.93325878898681877394087869495, −4.12358310244371971645360971100, −3.28080416789919277896388031589, −2.37933848390239807207610385109, −1.26117494844418294413087853899,
2.79313445964386687170906588628, 3.20299359259054561230143368856, 4.13049743937570584655639856599, 5.27093898878987406675062533346, 5.89448405261074710730969264743, 6.89558818049722505457601762518, 7.55405239464768541936686920241, 8.557047749408895434205516059415, 9.104016574272072670755266106052, 10.37890122810253048886547976188
