| L(s) = 1 | + (−8.27 − 7.71i)2-s + (45.6 − 18.9i)3-s + (8.86 + 127. i)4-s + (126. − 306. i)5-s + (−523. − 196. i)6-s + (1.15e3 + 1.15e3i)7-s + (912. − 1.12e3i)8-s + (182. − 182. i)9-s + (−3.41e3 + 1.55e3i)10-s + (4.80e3 + 1.99e3i)11-s + (2.82e3 + 5.66e3i)12-s + (−1.05e3 − 2.53e3i)13-s + (−642. − 1.85e4i)14-s − 1.64e4i·15-s + (−1.62e4 + 2.26e3i)16-s − 3.76e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.731 − 0.682i)2-s + (0.976 − 0.404i)3-s + (0.0692 + 0.997i)4-s + (0.454 − 1.09i)5-s + (−0.990 − 0.370i)6-s + (1.27 + 1.27i)7-s + (0.629 − 0.776i)8-s + (0.0833 − 0.0833i)9-s + (−1.08 + 0.491i)10-s + (1.08 + 0.450i)11-s + (0.471 + 0.946i)12-s + (−0.132 − 0.320i)13-s + (−0.0625 − 1.80i)14-s − 1.25i·15-s + (−0.990 + 0.138i)16-s − 1.85i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.65972 - 1.13696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65972 - 1.13696i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.27 + 7.71i)T \) |
| good | 3 | \( 1 + (-45.6 + 18.9i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-126. + 306. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-1.15e3 - 1.15e3i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-4.80e3 - 1.99e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.05e3 + 2.53e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 3.76e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (3.81e3 + 9.21e3i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-5.90e3 + 5.90e3i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-9.40e3 + 3.89e3i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.09e4 + 1.47e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (4.53e5 - 4.53e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-8.08e5 - 3.34e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 9.89e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (2.50e5 + 1.03e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (7.12e5 - 1.71e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.32e6 - 5.48e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.61e6 - 6.68e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.39e6 + 1.39e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.80e6 + 1.80e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 1.41e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-9.67e5 - 2.33e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-3.99e6 - 3.99e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.26e6T + 8.07e13T^{2} \) |
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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
−14.88279939352813972104352272312, −13.64167141856653477716306424674, −12.37311629347758781841344217767, −11.47592285824436180584870248699, −9.214712899466813403808086626078, −8.889500045813860823073331169165, −7.63687633636464755603052887978, −4.93692473224245814922545472259, −2.50016068554931353069546885819, −1.38797072416699629416372058523,
1.64652862359585292672466484963, 3.96726091552691983066514272940, 6.34084840806324438942460787294, 7.70245813556378576220959971671, 8.852572559571722838687032608388, 10.27799099673749254543124391322, 11.07996377557610658062886454500, 14.11531138512521961934965867129, 14.26991211233404669285693002357, 15.13741522168591944257937283427
