| L(s) = 1 | + 1.89i·3-s − 2·4-s − 0.651i·5-s − 2.29i·7-s − 0.608·9-s − 0.476i·11-s − 3.79i·12-s − 1.84·13-s + 1.23·15-s + 4·16-s + (4.10 − 0.412i)17-s + 2·19-s + 1.30i·20-s + 4.36·21-s − 7.91i·23-s + ⋯ |
| L(s) = 1 | + 1.09i·3-s − 4-s − 0.291i·5-s − 0.867i·7-s − 0.202·9-s − 0.143i·11-s − 1.09i·12-s − 0.511·13-s + 0.319·15-s + 16-s + (0.994 − 0.100i)17-s + 0.458·19-s + 0.291i·20-s + 0.951·21-s − 1.64i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20899 + 0.0606613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20899 + 0.0606613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.10 + 0.412i)T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 - 1.89iT - 3T^{2} \) |
| 5 | \( 1 + 0.651iT - 5T^{2} \) |
| 7 | \( 1 + 2.29iT - 7T^{2} \) |
| 11 | \( 1 + 0.476iT - 11T^{2} \) |
| 13 | \( 1 + 1.84T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 7.91iT - 23T^{2} \) |
| 29 | \( 1 - 1.30iT - 29T^{2} \) |
| 31 | \( 1 - 4.62iT - 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 1.59iT - 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 6.34iT - 71T^{2} \) |
| 73 | \( 1 + 6.18iT - 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 9.21iT - 97T^{2} \) |
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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
−10.42013414152195779959463967715, −9.542410597711042757257173157736, −8.961020290014797285425329172058, −7.985269953443326769344350548441, −7.01659443948993204898466303914, −5.54517533393196766165912991826, −4.74348722410126861873270566426, −4.14867361450338764781090371608, −3.16520281289701695280289998812, −0.845234776717766790714202691061,
1.13343606977551366554098428363, 2.54774941704076361299157788527, 3.80421918600617896495522112587, 5.19118738453823479222163145928, 5.82289592312001602527171104680, 7.06330882576869259067993554772, 7.73815772405463640387269654699, 8.565164750793499610559466255521, 9.524799436061482618551086582085, 10.11154701306528187874199083445
