| 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
−11.13829629426406517603614421966, −10.34930191423673614644003991478, −10.03408495183803332652858982275, −8.668188620530832832367982036925, −7.937560640792895917743199480439, −6.59557267961808553953777448567, −6.10747883635836062793714446255, −4.85105363344463217442469085628, −3.55010095059108480221248772177, −1.69058232150181016088642182537,
0.57147598044164860495339141134, 2.21792565094453968856473821622, 3.66310461763277189275222430639, 5.25795512923730758427291649872, 6.36863530295795380662582712498, 6.87424011517095237600075017708, 8.478136195214044491007039417444, 8.993536463931282299626205606284, 9.813965692969240481261200374838, 11.06747945232154083079099772081
