| L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.993 − 0.677i)3-s + (0.623 + 0.781i)4-s + (0.733 − 0.680i)5-s + (0.601 + 1.04i)6-s + (−0.895 + 1.55i)7-s + (−0.222 − 0.974i)8-s + (−0.567 − 1.44i)9-s + (−0.955 + 0.294i)10-s + (1.01 − 1.26i)11-s + (−0.0898 − 1.19i)12-s + (3.87 + 1.19i)13-s + (1.47 − 1.00i)14-s + (−1.18 + 0.179i)15-s + (−0.222 + 0.974i)16-s + (−4.64 − 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.573 − 0.391i)3-s + (0.311 + 0.390i)4-s + (0.327 − 0.304i)5-s + (0.245 + 0.425i)6-s + (−0.338 + 0.586i)7-s + (−0.0786 − 0.344i)8-s + (−0.189 − 0.482i)9-s + (−0.302 + 0.0932i)10-s + (0.304 − 0.382i)11-s + (−0.0259 − 0.346i)12-s + (1.07 + 0.331i)13-s + (0.395 − 0.269i)14-s + (−0.306 + 0.0462i)15-s + (−0.0556 + 0.243i)16-s + (−1.12 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.314199 - 0.599142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.314199 - 0.599142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 43 | \( 1 + (6.48 + 0.964i)T \) |
| good | 3 | \( 1 + (0.993 + 0.677i)T + (1.09 + 2.79i)T^{2} \) |
| 7 | \( 1 + (0.895 - 1.55i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 1.26i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.87 - 1.19i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (4.64 + 4.30i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.64 + 4.19i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (4.73 + 0.713i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.04 + 3.43i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.0732 + 0.977i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.81 + 2.80i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (7.18 + 9.01i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-7.40 + 2.28i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.341 - 1.49i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.840 + 11.2i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (4.98 - 12.7i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 1.87i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-14.1 - 4.37i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (3.15 - 5.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.48 - 6.46i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-3.21 - 2.19i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (0.162 - 0.203i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−11.06747945232154083079099772081, −9.813965692969240481261200374838, −8.993536463931282299626205606284, −8.478136195214044491007039417444, −6.87424011517095237600075017708, −6.36863530295795380662582712498, −5.25795512923730758427291649872, −3.66310461763277189275222430639, −2.21792565094453968856473821622, −0.57147598044164860495339141134,
1.69058232150181016088642182537, 3.55010095059108480221248772177, 4.85105363344463217442469085628, 6.10747883635836062793714446255, 6.59557267961808553953777448567, 7.937560640792895917743199480439, 8.668188620530832832367982036925, 10.03408495183803332652858982275, 10.34930191423673614644003991478, 11.13829629426406517603614421966
