| L(s) = 1 | + (7.93 − 13.7i)5-s + (7.93 + 13.7i)11-s + 26·13-s + (−39.6 − 68.7i)17-s + (−34 + 58.8i)19-s + (23.8 − 41.2i)23-s + (−63.5 − 109. i)25-s + 253.·29-s + (−106 − 183. i)31-s + (−109 + 188. i)37-s − 396.·41-s + 260·43-s + (206. − 357. i)47-s + (−238. − 412. i)53-s + 252·55-s + ⋯ |
| L(s) = 1 | + (0.709 − 1.22i)5-s + (0.217 + 0.376i)11-s + 0.554·13-s + (−0.566 − 0.980i)17-s + (−0.410 + 0.711i)19-s + (0.215 − 0.373i)23-s + (−0.508 − 0.879i)25-s + 1.62·29-s + (−0.614 − 1.06i)31-s + (−0.484 + 0.838i)37-s − 1.51·41-s + 0.922·43-s + (0.640 − 1.10i)47-s + (−0.617 − 1.06i)53-s + 0.617·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.988769560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988769560\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-7.93 + 13.7i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.93 - 13.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 26T + 2.19e3T^{2} \) |
| 17 | \( 1 + (39.6 + 68.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (34 - 58.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-23.8 + 41.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (106 + 183. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (109 - 188. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-206. + 357. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (238. + 412. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (142. + 247. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-161 + 278. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (178 + 308. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-113 - 195. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (220 - 381. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-103. + 178. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−8.630888996559014389027070873089, −8.172982936258391272194249122369, −6.92211242074298463264052321937, −6.25228456213774353295466244111, −5.24169616775711685949650710337, −4.72119162388766358636475585407, −3.71711629805521054744976867424, −2.36102949601614990859665008211, −1.45509538617499389447480383563, −0.41879116089297839315015161529,
1.26192970783408412628407009884, 2.36978066036848677095724844974, 3.17843069620267052087326630378, 4.12456863638752191576269215668, 5.29524071945296012690776808714, 6.29694916414721993012213818685, 6.58271843551666450876949848066, 7.50230698055538417304384181744, 8.604676416704166700866973293836, 9.103462927790234235897909923075
