| L(s) = 1 | + (−1.31 + 4.06i)2-s + (−8.28 − 6.02i)4-s + (−4.79 + 10.0i)5-s + 33.6·7-s + (7.75 − 5.63i)8-s + (−34.6 − 32.8i)10-s + (−9.92 + 30.5i)11-s + (23.5 + 72.5i)13-s + (−44.3 + 136. i)14-s + (−12.6 − 39.0i)16-s + (−33.4 + 24.3i)17-s + (−2.91 + 2.11i)19-s + (100. − 54.7i)20-s + (−110. − 80.6i)22-s + (38.2 − 117. i)23-s + ⋯ |
| L(s) = 1 | + (−0.466 + 1.43i)2-s + (−1.03 − 0.752i)4-s + (−0.429 + 0.903i)5-s + 1.81·7-s + (0.342 − 0.248i)8-s + (−1.09 − 1.03i)10-s + (−0.272 + 0.837i)11-s + (0.502 + 1.54i)13-s + (−0.846 + 2.60i)14-s + (−0.198 − 0.609i)16-s + (−0.477 + 0.347i)17-s + (−0.0351 + 0.0255i)19-s + (1.12 − 0.612i)20-s + (−1.07 − 0.781i)22-s + (0.346 − 1.06i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.361804 - 1.12787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.361804 - 1.12787i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (4.79 - 10.0i)T \) |
| good | 2 | \( 1 + (1.31 - 4.06i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 - 33.6T + 343T^{2} \) |
| 11 | \( 1 + (9.92 - 30.5i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-23.5 - 72.5i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (33.4 - 24.3i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (2.91 - 2.11i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-38.2 + 117. i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-153. - 111. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (267. - 194. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (39.9 + 122. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (129. + 398. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 41.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (375. + 272. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-200. - 145. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-66.6 - 205. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-57.9 + 178. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (223. - 162. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-328. - 238. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (161. - 496. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-295. - 214. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (550. - 399. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (91.6 - 281. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (185. + 135. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−12.19061183984896082105524111067, −11.26004584323619050649282648009, −10.47254083227085914805032531226, −8.879407908383563653328478961347, −8.332241560802477629352084780354, −7.16511513224534975627314637345, −6.79076183465503486750684563817, −5.24606580317428191540261220138, −4.26139262808131923963725904398, −1.98428333259904876315505569709,
0.60066484065873168009568541928, 1.62852676650285203522462734411, 3.23212973837879531259049292234, 4.55864284855634613730791337257, 5.59332222411875089124953816056, 7.954792811723474861236271518487, 8.273826352048820049380069395504, 9.322741810557718436926528883886, 10.57092881488647435518085147284, 11.36570134679369905366777270415
