L(s) = 1 | + (−0.0347 − 0.106i)2-s + (−0.809 − 0.587i)3-s + (1.60 − 1.16i)4-s + (0.327 − 1.00i)5-s + (−0.0347 + 0.106i)6-s + (0.809 − 0.587i)7-s + (−0.362 − 0.263i)8-s + (0.309 + 0.951i)9-s − 0.119·10-s + (−1.98 + 2.65i)11-s − 1.98·12-s + (−1.80 − 5.55i)13-s + (−0.0909 − 0.0661i)14-s + (−0.858 + 0.623i)15-s + (1.21 − 3.73i)16-s + (0.866 − 2.66i)17-s + ⋯ |
L(s) = 1 | + (−0.0245 − 0.0756i)2-s + (−0.467 − 0.339i)3-s + (0.803 − 0.584i)4-s + (0.146 − 0.451i)5-s + (−0.0141 + 0.0436i)6-s + (0.305 − 0.222i)7-s + (−0.128 − 0.0932i)8-s + (0.103 + 0.317i)9-s − 0.0377·10-s + (−0.597 + 0.801i)11-s − 0.573·12-s + (−0.500 − 1.54i)13-s + (−0.0243 − 0.0176i)14-s + (−0.221 + 0.161i)15-s + (0.303 − 0.933i)16-s + (0.210 − 0.647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.987789 - 0.762427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.987789 - 0.762427i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (1.98 - 2.65i)T \) |
good | 2 | \( 1 + (0.0347 + 0.106i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.327 + 1.00i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (1.80 + 5.55i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 2.66i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.88 - 2.09i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 + (6.95 - 5.05i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.538 - 1.65i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.83 - 2.78i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.96 + 3.60i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 + (-6.62 - 4.81i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.43 - 13.6i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.15 - 2.29i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.55 - 14.0i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + (-3.59 + 11.0i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.39 + 2.46i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.10 - 6.47i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.360 + 1.10i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 + (1.49 + 4.61i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.13583741973218977773242674948, −10.86350884054011338823955558165, −10.38314489231897087577319122095, −9.251540030733974633615175671530, −7.61599504832691758595523392036, −7.15657374619256554930073410363, −5.53942967214852638311793446442, −5.10663490406368196458413601257, −2.84286617326086480018842749357, −1.21318262706891756834934299029,
2.29709192346879545223094572797, 3.68955542039957253961553154446, 5.22035119039282280749299462413, 6.41045382618271855312901183260, 7.23681297898121849805441787779, 8.413190818424927074868004592357, 9.547438788016283934502237970915, 10.83190342747959610824823246013, 11.32264603119056158233719431091, 12.13182020799164244401410245851