| L(s) = 1 | + (−2.71 + 2.71i)2-s + (−16.9 − 16.9i)3-s + 49.2i·4-s + 92.2·6-s + (383. − 383. i)7-s + (−307. − 307. i)8-s − 152. i·9-s + 2.40e3·11-s + (836. − 836. i)12-s + (−332. − 332. i)13-s + 2.08e3i·14-s − 1.48e3·16-s + (−373. + 373. i)17-s + (412. + 412. i)18-s − 7.03e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.339 + 0.339i)2-s + (−0.629 − 0.629i)3-s + 0.769i·4-s + 0.427·6-s + (1.11 − 1.11i)7-s + (−0.600 − 0.600i)8-s − 0.208i·9-s + 1.80·11-s + (0.484 − 0.484i)12-s + (−0.151 − 0.151i)13-s + 0.759i·14-s − 0.361·16-s + (−0.0760 + 0.0760i)17-s + (0.0707 + 0.0707i)18-s − 1.02i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.10167 - 0.373816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10167 - 0.373816i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (2.71 - 2.71i)T - 64iT^{2} \) |
| 3 | \( 1 + (16.9 + 16.9i)T + 729iT^{2} \) |
| 7 | \( 1 + (-383. + 383. i)T - 1.17e5iT^{2} \) |
| 11 | \( 1 - 2.40e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (332. + 332. i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + (373. - 373. i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + 7.03e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-2.61e3 - 2.61e3i)T + 1.48e8iT^{2} \) |
| 29 | \( 1 + 1.36e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.07e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (-2.99e4 + 2.99e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 6.03e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (6.38e4 + 6.38e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + (7.43e4 - 7.43e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 + (-1.25e5 - 1.25e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 - 1.52e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (6.90e4 - 6.90e4i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + 9.61e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (2.31e5 + 2.31e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + 2.81e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.49e5 - 5.49e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 1.19e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (4.69e5 - 4.69e5i)T - 8.32e11iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.74006698737796375145374996218, −14.90804558208394689299135738592, −13.53542199092742777403764188744, −12.08611379143336515825086115115, −11.24946675573742287203436923790, −9.123568970711208822066572094911, −7.55371537600662589247914850601, −6.58579002784176775462681815078, −4.12027840476581552872692332130, −0.950815043846376699739184034363,
1.67056072856175309878265312608, 4.77011644452867321874158136784, 6.04815534560183398854495207861, 8.622815377776185976105994374416, 9.846157876816497862757409127095, 11.26515523552751942787209026212, 11.86570791677709127206121665799, 14.33665464616003506014851481398, 15.01286767457983815488800677627, 16.51557873147006557118002439362
