| L(s) = 1 | + (0.723 − 1.25i)5-s + (23.0 + 39.9i)11-s + 32.2·13-s + (−38.8 − 67.3i)17-s + (6.33 − 10.9i)19-s + (−50.4 + 87.4i)23-s + (61.4 + 106. i)25-s − 213.·29-s + (21.0 + 36.4i)31-s + (−155. + 268. i)37-s + 44.0·41-s + 381.·43-s + (179. − 310. i)47-s + (92.4 + 160. i)53-s + 66.7·55-s + ⋯ |
| L(s) = 1 | + (0.0646 − 0.112i)5-s + (0.632 + 1.09i)11-s + 0.687·13-s + (−0.554 − 0.961i)17-s + (0.0764 − 0.132i)19-s + (−0.457 + 0.792i)23-s + (0.491 + 0.851i)25-s − 1.36·29-s + (0.121 + 0.211i)31-s + (−0.688 + 1.19i)37-s + 0.167·41-s + 1.35·43-s + (0.555 − 0.962i)47-s + (0.239 + 0.415i)53-s + 0.163·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.435368824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435368824\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-0.723 + 1.25i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-23.0 - 39.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (38.8 + 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6.33 + 10.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (50.4 - 87.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-21.0 - 36.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (155. - 268. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 44.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-179. + 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-92.4 - 160. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (227. + 393. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-5.92 + 10.2i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (295. + 511. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-487. - 844. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (149. - 259. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-347. + 602. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
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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
−9.286191399933692902672426712062, −8.519582711092315874646341110558, −7.38457584173001189966099117376, −6.99564475211086126326117061105, −5.95065663107156482050883135034, −5.09018147921569842974163051985, −4.22787554441506445415742962051, −3.33795765359656333893010673073, −2.10357420487986552593820261869, −1.17828737200704121073959289022,
0.31492021047293474668080309868, 1.48344129938088278693947755420, 2.62938410518615903235292227433, 3.76251976209468732228314799642, 4.31267797096039500527496983914, 5.80475584750062835761035579293, 6.06635289375375711430403467263, 7.04745438349592923699760889065, 8.028638220705643095859986838859, 8.763003181805660484891331639127
