| L(s) = 1 | + (0.108 − 0.334i)2-s + (1.93 − 2.15i)3-s + (1.51 + 1.10i)4-s + (−1.48 + 2.57i)5-s + (−0.508 − 0.880i)6-s + (0.113 − 1.07i)7-s + (1.10 − 0.800i)8-s + (−0.561 − 5.34i)9-s + (0.699 + 0.776i)10-s + (2.22 − 0.988i)11-s + (5.31 − 1.12i)12-s + (2.81 + 0.597i)13-s + (−0.347 − 0.154i)14-s + (2.65 + 8.18i)15-s + (1.01 + 3.11i)16-s + (1.66 + 0.742i)17-s + ⋯ |
| L(s) = 1 | + (0.0767 − 0.236i)2-s + (1.11 − 1.24i)3-s + (0.759 + 0.551i)4-s + (−0.664 + 1.15i)5-s + (−0.207 − 0.359i)6-s + (0.0427 − 0.406i)7-s + (0.389 − 0.283i)8-s + (−0.187 − 1.78i)9-s + (0.221 + 0.245i)10-s + (0.669 − 0.298i)11-s + (1.53 − 0.325i)12-s + (0.779 + 0.165i)13-s + (−0.0928 − 0.0413i)14-s + (0.686 + 2.11i)15-s + (0.252 + 0.778i)16-s + (0.404 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.60378 - 0.969765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60378 - 0.969765i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.108 + 0.334i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.93 + 2.15i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (1.48 - 2.57i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.113 + 1.07i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.22 + 0.988i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.81 - 0.597i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.66 - 0.742i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (2.07 - 0.440i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.357 - 0.259i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.976 + 3.00i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.57 - 2.72i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.64 + 5.16i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (8.22 - 1.74i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (2.46 + 7.57i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.520 + 4.94i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (8.03 - 8.92i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + (-3.21 + 5.57i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.175 + 1.66i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (13.1 - 5.84i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-3.14 - 1.40i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (8.60 + 9.55i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.82 - 1.32i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.76 + 2.00i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.07126989518946104905546378849, −8.638731957207118903751847885290, −8.162120771563825565758018232296, −7.32456655312613609244220277728, −6.83340236280844135031750275296, −6.21931439018496710813741274537, −3.86432330383862070867416510173, −3.43363923091549023957672922198, −2.50069198562671208188312319901, −1.42781109226588996945734676493,
1.49247343118003240854163686680, 2.88485152401912095559080736505, 3.94994228180939266050854039544, 4.72677188784534079702798840931, 5.54147546594502294732520372900, 6.75002913709363327503155120292, 7.987264280640594294246481330526, 8.491290428968945658802065921470, 9.234136816474343564775720922549, 9.930970307359384825877831798140
