L(s) = 1 | + (−1.25 + 0.643i)2-s + (0.222 + 0.974i)3-s + (1.17 − 1.62i)4-s + (0.631 − 0.144i)5-s + (−0.907 − 1.08i)6-s + (−2.32 − 1.26i)7-s + (−0.434 + 2.79i)8-s + (−0.900 + 0.433i)9-s + (−0.703 + 0.588i)10-s + (2.50 − 5.19i)11-s + (1.84 + 0.782i)12-s + (−2.15 + 4.47i)13-s + (3.74 + 0.102i)14-s + (0.281 + 0.584i)15-s + (−1.25 − 3.79i)16-s + (6.08 + 4.84i)17-s + ⋯ |
L(s) = 1 | + (−0.890 + 0.454i)2-s + (0.128 + 0.562i)3-s + (0.586 − 0.810i)4-s + (0.282 − 0.0645i)5-s + (−0.370 − 0.442i)6-s + (−0.877 − 0.479i)7-s + (−0.153 + 0.988i)8-s + (−0.300 + 0.144i)9-s + (−0.222 + 0.186i)10-s + (0.754 − 1.56i)11-s + (0.531 + 0.225i)12-s + (−0.598 + 1.24i)13-s + (0.999 + 0.0273i)14-s + (0.0726 + 0.150i)15-s + (−0.312 − 0.949i)16-s + (1.47 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.970982 + 0.344851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.970982 + 0.344851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.25 - 0.643i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (2.32 + 1.26i)T \) |
good | 5 | \( 1 + (-0.631 + 0.144i)T + (4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-2.50 + 5.19i)T + (-6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (2.15 - 4.47i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-6.08 - 4.84i)T + (3.78 + 16.5i)T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 + (-2.05 + 1.64i)T + (5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-4.35 + 5.46i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 - 4.28T + 31T^{2} \) |
| 37 | \( 1 + (-1.37 + 1.72i)T + (-8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + (-7.63 + 1.74i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.773 - 0.176i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (0.374 + 0.180i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-2.35 - 2.95i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (1.21 - 5.32i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (3.31 + 2.64i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + (3.04 - 2.42i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (2.91 + 6.05i)T + (-45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 - 2.62iT - 79T^{2} \) |
| 83 | \( 1 + (-5.45 + 2.62i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 6.28i)T + (-55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + 2.58iT - 97T^{2} \) |
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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
−10.45622033734792031366697322189, −9.712150706531647599655139122585, −9.241743767581838032109657434101, −8.274401580093200129905879051697, −7.31387393699136726920640557659, −6.20749516229960874652021955313, −5.69718523830017671184514072722, −4.09294942324127546398182296070, −2.94473354450167219900903319410, −1.06846100148440019765285071936,
1.05996602038489807953139869971, 2.57062341158149893183415501535, 3.30930595120400625931088032941, 5.14279494704190865962887506161, 6.36608643930829963169632692946, 7.34601194434902356032997982259, 7.76365983036909100237037181470, 9.163586479994357968877425778758, 9.757725561483422414237771092002, 10.15430424352879898784059685888