| 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
−9.930970307359384825877831798140, −9.234136816474343564775720922549, −8.491290428968945658802065921470, −7.987264280640594294246481330526, −6.75002913709363327503155120292, −5.54147546594502294732520372900, −4.72677188784534079702798840931, −3.94994228180939266050854039544, −2.88485152401912095559080736505, −1.49247343118003240854163686680,
1.42781109226588996945734676493, 2.50069198562671208188312319901, 3.43363923091549023957672922198, 3.86432330383862070867416510173, 6.21931439018496710813741274537, 6.83340236280844135031750275296, 7.32456655312613609244220277728, 8.162120771563825565758018232296, 8.638731957207118903751847885290, 10.07126989518946104905546378849
