| L(s) = 1 | + (1.41 + 1.41i)2-s + 4.00i·4-s + (−9 + 9i)7-s + (−5.65 + 5.65i)8-s + 38.1i·11-s + (9 + 9i)13-s − 25.4·14-s − 16.0·16-s + (−55.1 − 55.1i)17-s − 38i·19-s + (−54.0 + 54.0i)22-s + (29.6 − 29.6i)23-s + 25.4i·26-s + (−36.0 − 36.0i)28-s − 76.3·29-s + ⋯ |
| L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.485 + 0.485i)7-s + (−0.250 + 0.250i)8-s + 1.04i·11-s + (0.192 + 0.192i)13-s − 0.485·14-s − 0.250·16-s + (−0.786 − 0.786i)17-s − 0.458i·19-s + (−0.523 + 0.523i)22-s + (0.269 − 0.269i)23-s + 0.192i·26-s + (−0.242 − 0.242i)28-s − 0.489·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7853099853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7853099853\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 1.41i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (9 - 9i)T - 343iT^{2} \) |
| 11 | \( 1 - 38.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-9 - 9i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (55.1 + 55.1i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 38iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-29.6 + 29.6i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 76.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 236T + 2.97e4T^{2} \) |
| 37 | \( 1 + (279 - 279i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 343. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (360 + 360i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (254. + 254. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-352. + 352. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 110T + 2.26e5T^{2} \) |
| 67 | \( 1 + (702 - 702i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 76.3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (378 + 378i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 272iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (59.3 - 59.3i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (684 - 684i)T - 9.12e5iT^{2} \) |
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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
−11.43169332868831801302788431092, −10.18536453893052465522018873052, −9.251171309144406393787004462798, −8.489219021824306659191291016564, −7.14267238588512897097662773666, −6.70057991276880334571534958722, −5.41611405860576637532098272494, −4.59633212397918176774455916260, −3.32873846847170543318360745956, −2.06963226212020271946855033376,
0.19846036180063258910948768049, 1.74462721346389598649262197041, 3.28813772196762126830788766029, 3.97092990057339165679112267826, 5.38326080646385745449066775915, 6.21509301682748644268187553463, 7.26220973281247833834027090538, 8.510236042276687160167996973890, 9.344173272203479787107494526656, 10.52763136747876610523983156541
