| L(s) = 1 | + (2.16 − 1.33i)2-s + (−0.273 − 0.961i)3-s + (1.98 − 3.99i)4-s + (−1.14 + 0.442i)5-s + (−1.87 − 1.71i)6-s + (−0.0140 + 0.151i)7-s + (−0.577 − 6.23i)8-s + (−0.850 + 0.526i)9-s + (−1.87 + 2.48i)10-s + (2.29 − 1.42i)11-s + (−4.38 − 0.820i)12-s + (−0.118 + 1.27i)13-s + (0.171 + 0.345i)14-s + (0.738 + 0.977i)15-s + (−4.21 − 5.57i)16-s + (−0.583 + 0.531i)17-s + ⋯ |
| L(s) = 1 | + (1.52 − 0.946i)2-s + (−0.157 − 0.555i)3-s + (0.994 − 1.99i)4-s + (−0.511 + 0.197i)5-s + (−0.766 − 0.699i)6-s + (−0.00529 + 0.0571i)7-s + (−0.204 − 2.20i)8-s + (−0.283 + 0.175i)9-s + (−0.593 + 0.786i)10-s + (0.691 − 0.428i)11-s + (−1.26 − 0.236i)12-s + (−0.0328 + 0.354i)13-s + (0.0459 + 0.0923i)14-s + (0.190 + 0.252i)15-s + (−1.05 − 1.39i)16-s + (−0.141 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43989 - 2.14233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43989 - 2.14233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.273 + 0.961i)T \) |
| 103 | \( 1 + (6.57 + 7.73i)T \) |
| good | 2 | \( 1 + (-2.16 + 1.33i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (1.14 - 0.442i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.0140 - 0.151i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.42i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.118 - 1.27i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (0.583 - 0.531i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.297 - 1.04i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (-3.93 - 2.43i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-1.74 + 0.676i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-2.71 - 3.59i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 1.21i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (11.0 + 4.27i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (6.98 - 1.30i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 0.266T + 47T^{2} \) |
| 53 | \( 1 + (-3.37 + 11.8i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.438 + 4.73i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (8.00 - 7.29i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.590 - 6.37i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (10.5 + 4.09i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (9.47 + 3.67i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (4.68 - 1.81i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (1.09 - 11.7i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-0.787 - 1.58i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-6.38 - 5.82i)T + (8.95 + 96.5i)T^{2} \) |
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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
−11.70446376259977828954911509029, −11.02785971814135288249393407189, −9.964238781487182419259293588079, −8.575043403170453354929786796741, −7.10636502291153226452255617387, −6.23904570595180157946551000891, −5.17780328727797826792139747142, −4.00243069390587406357484800168, −3.02907452887112469825969805763, −1.53291094884410153669367157200,
2.98989586401755205938590230081, 4.19212020546963112549327429485, 4.76324579296670611838083611735, 5.94747066208598113159599608799, 6.85038249756866092994448496908, 7.84404939602615304671982371412, 8.900908220452927208270044365969, 10.26442233114256671885925025221, 11.57763057245655975829718852264, 12.05122890752304595559222053589
