| L(s) = 1 | + (10.7 + 3.49i)2-s + (−23.2 + 16.8i)3-s + (103. + 75.2i)4-s + (88.9 + 273. i)5-s + (−308. + 100. i)6-s + (−550. + 757. i)7-s + (851. + 1.17e3i)8-s + (−1.77e3 + 5.45e3i)9-s + 3.25e3i·10-s + (−4.52e3 + 1.39e4i)11-s − 3.67e3·12-s + (2.68e3 + 870. i)13-s + (−8.56e3 + 6.22e3i)14-s + (−6.68e3 − 4.85e3i)15-s + (5.06e3 + 1.55e4i)16-s + (758. − 246. i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.286 + 0.208i)3-s + (0.404 + 0.293i)4-s + (0.142 + 0.438i)5-s + (−0.238 + 0.0773i)6-s + (−0.229 + 0.315i)7-s + (0.207 + 0.286i)8-s + (−0.270 + 0.831i)9-s + 0.325i·10-s + (−0.308 + 0.951i)11-s − 0.177·12-s + (0.0938 + 0.0304i)13-s + (−0.223 + 0.162i)14-s + (−0.132 − 0.0959i)15-s + (0.0772 + 0.237i)16-s + (0.00908 − 0.00295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.20291 + 1.61818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20291 + 1.61818i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.7 - 3.49i)T \) |
| 11 | \( 1 + (4.52e3 - 1.39e4i)T \) |
| good | 3 | \( 1 + (23.2 - 16.8i)T + (2.02e3 - 6.23e3i)T^{2} \) |
| 5 | \( 1 + (-88.9 - 273. i)T + (-3.16e5 + 2.29e5i)T^{2} \) |
| 7 | \( 1 + (550. - 757. i)T + (-1.78e6 - 5.48e6i)T^{2} \) |
| 13 | \( 1 + (-2.68e3 - 870. i)T + (6.59e8 + 4.79e8i)T^{2} \) |
| 17 | \( 1 + (-758. + 246. i)T + (5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-3.51e4 - 4.83e4i)T + (-5.24e9 + 1.61e10i)T^{2} \) |
| 23 | \( 1 - 3.81e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-4.98e5 + 6.85e5i)T + (-1.54e11 - 4.75e11i)T^{2} \) |
| 31 | \( 1 + (-3.76e5 + 1.15e6i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.98e4 - 1.44e4i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-1.17e6 - 1.62e6i)T + (-2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 3.69e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (2.22e6 - 1.61e6i)T + (7.35e12 - 2.26e13i)T^{2} \) |
| 53 | \( 1 + (2.79e5 - 8.61e5i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (9.84e6 + 7.15e6i)T + (4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-9.16e6 + 2.97e6i)T + (1.55e14 - 1.12e14i)T^{2} \) |
| 67 | \( 1 - 6.36e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-9.27e6 - 2.85e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-1.66e7 + 2.28e7i)T + (-2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (1.16e7 + 3.77e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.64e7 - 5.34e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 - 8.57e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.75e6 + 1.77e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−16.30817713153787812727653843198, −15.21930872213689026397877950142, −14.00897081650186098787623183126, −12.71477901461899042649456845219, −11.33223925859471521152949815663, −9.972543637328463134938651977058, −7.83468155670665613572095308562, −6.18378233804484372990319651818, −4.66684552657929963972567301678, −2.53096092941584782896766049853,
0.857523970132911197569898031169, 3.31609863992194972641441929108, 5.29900055413608103556993734901, 6.74891907310513605539967258146, 8.864646417749156825989299833000, 10.65190257986363387363077147364, 11.98603697481724259035852096621, 13.10012237378109480074816730571, 14.25329884735961205197190783643, 15.73976767267569805188695488865
