| L(s) = 1 | + (−4.5 + 7.79i)3-s + (−17.6 − 30.6i)5-s + (−40.5 − 70.1i)9-s + (−21.7 + 37.7i)11-s − 648.·13-s + 318.·15-s + (299. − 518. i)17-s + (−320. − 555. i)19-s + (135. + 235. i)23-s + (936. − 1.62e3i)25-s + 729·27-s − 5.28e3·29-s + (−221. + 383. i)31-s + (−196. − 339. i)33-s + (4.44e3 + 7.69e3i)37-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.316 − 0.548i)5-s + (−0.166 − 0.288i)9-s + (−0.0542 + 0.0939i)11-s − 1.06·13-s + 0.365·15-s + (0.251 − 0.435i)17-s + (−0.203 − 0.352i)19-s + (0.0535 + 0.0927i)23-s + (0.299 − 0.519i)25-s + 0.192·27-s − 1.16·29-s + (−0.0414 + 0.0717i)31-s + (−0.0313 − 0.0542i)33-s + (0.533 + 0.924i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9212580648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9212580648\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (17.6 + 30.6i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (21.7 - 37.7i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 648.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-299. + 518. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (320. + 555. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-135. - 235. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (221. - 383. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-4.44e3 - 7.69e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 7.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.50e3 + 2.61e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.55e3 + 6.16e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.64e4 + 2.84e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (9.58e3 + 1.65e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (7.49e3 - 1.29e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-5.74e3 + 9.94e3i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.70e4 - 4.67e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.45e4 - 7.71e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 3.58e4T + 8.58e9T^{2} \) |
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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
−9.983434871023655906683557544157, −9.375801446252779900148752739949, −8.403010523582161637776921814206, −7.48604831012962188112347549751, −6.49158636155411251435873686893, −5.23641202956109717241233997441, −4.69844258414365679295064646166, −3.57746665008316057433151674246, −2.30861739763487503342694953904, −0.74983469601561607380404949101,
0.29162224823871025266661263449, 1.73735678507461767663967026368, 2.85050626624872558284038439762, 4.00593786637996127065493749911, 5.24048830953022228231909458603, 6.13326702175902869832949516158, 7.21185261103063538891272431401, 7.65380014309563899315013086958, 8.793290123791262103038151508258, 9.824257090554753333401471114441
