| L(s) = 1 | + (−1.68 − 1.22i)2-s + (0.111 + 1.06i)3-s + (0.717 + 2.20i)4-s + (−0.5 − 0.866i)5-s + (1.11 − 1.92i)6-s + (0.859 + 0.182i)7-s + (0.206 − 0.634i)8-s + (1.81 − 0.385i)9-s + (−0.217 + 2.06i)10-s + (3.40 − 3.77i)11-s + (−2.27 + 1.01i)12-s + (2.24 + 0.998i)13-s + (−1.22 − 1.35i)14-s + (0.866 − 0.629i)15-s + (2.63 − 1.91i)16-s + (−1.92 − 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.863i)2-s + (0.0646 + 0.615i)3-s + (0.358 + 1.10i)4-s + (−0.223 − 0.387i)5-s + (0.454 − 0.787i)6-s + (0.324 + 0.0690i)7-s + (0.0729 − 0.224i)8-s + (0.604 − 0.128i)9-s + (−0.0687 + 0.653i)10-s + (1.02 − 1.13i)11-s + (−0.655 + 0.291i)12-s + (0.621 + 0.276i)13-s + (−0.326 − 0.362i)14-s + (0.223 − 0.162i)15-s + (0.658 − 0.478i)16-s + (−0.467 − 0.519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620943 - 0.310389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620943 - 0.310389i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-3.34 - 4.45i)T \) |
| good | 2 | \( 1 + (1.68 + 1.22i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.111 - 1.06i)T + (-2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (-0.859 - 0.182i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-3.40 + 3.77i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.24 - 0.998i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (1.92 + 2.14i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.240 - 0.106i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.30 + 4.00i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.229 - 0.166i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.18 - 3.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.15 - 10.9i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-1.59 + 0.708i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-3.95 + 2.87i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (9.05 - 1.92i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.18 + 11.3i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + (-0.0585 - 0.101i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.97 - 1.48i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.05 + 6.72i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (6.64 + 7.37i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.15 - 10.9i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-3.60 - 11.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.44 - 4.43i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.36459727894962199907986125481, −11.41627033318116865708130502269, −10.79988808691506743901322459487, −9.664962765215785261649269324441, −8.904818984594891416129321595258, −8.210529937784490711787454138638, −6.54498644685963734411646772998, −4.71804829068045284975499769303, −3.32758441968863499792117844898, −1.26920994959177364763337512314,
1.56866542645515587729681492877, 4.12095882228862258476581687214, 6.15240287336980543357151486690, 7.09240905030224946138072210043, 7.68370326962478908207489787910, 8.814973937085219108387034122416, 9.800014415341675992082448533236, 10.81013991827403319700542171490, 12.10572766164688030737546376015, 13.08527133418097801802252403050
