L(s) = 1 | − 1.80i·2-s + 3-s − 1.24·4-s − 1.44i·5-s − 1.80i·6-s + 3.44i·7-s − 1.35i·8-s + 9-s − 2.60·10-s − 5.18i·11-s − 1.24·12-s + 6.20·14-s − 1.44i·15-s − 4.93·16-s + 0.753·17-s − 1.80i·18-s + ⋯ |
L(s) = 1 | − 1.27i·2-s + 0.577·3-s − 0.623·4-s − 0.646i·5-s − 0.735i·6-s + 1.30i·7-s − 0.479i·8-s + 0.333·9-s − 0.823·10-s − 1.56i·11-s − 0.359·12-s + 1.65·14-s − 0.373i·15-s − 1.23·16-s + 0.182·17-s − 0.424i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.690366 - 1.61716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.690366 - 1.61716i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.80iT - 2T^{2} \) |
| 5 | \( 1 + 1.44iT - 5T^{2} \) |
| 7 | \( 1 - 3.44iT - 7T^{2} \) |
| 11 | \( 1 + 5.18iT - 11T^{2} \) |
| 17 | \( 1 - 0.753T + 17T^{2} \) |
| 19 | \( 1 + 7.96iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 + 1.80iT - 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 - 1.87iT - 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 - 4.54iT - 67T^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 2.95iT - 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 - 6.46iT - 83T^{2} \) |
| 89 | \( 1 - 1.15iT - 89T^{2} \) |
| 97 | \( 1 + 8.65iT - 97T^{2} \) |
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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
−10.89570649854550246496585626676, −9.553910303922076851578856017307, −8.918776482829683718927052631117, −8.486664999424894007055833509166, −6.94586781274760156417771851723, −5.69319865909817833229851154249, −4.60814935532054668690156607025, −3.16978837543777033337460658369, −2.60781302073818172050014928076, −1.06490786971069116226107588199,
2.01890488757516149850030633341, 3.67340077188047881825447126990, 4.63888975635074699043875745028, 5.94168052563671430389541916584, 7.10503109826280366554765691167, 7.33181856132040904951112933473, 8.121825428960928016400926081703, 9.394130547054549034663465111676, 10.21498434275851988567688332437, 10.96872732954285040827843857846