| L(s) = 1 | + (0.0936 − 0.128i)2-s + (−0.602 + 1.85i)3-s + (1.22 + 3.78i)4-s + (−1.80 + 1.31i)5-s + (0.182 + 0.251i)6-s + (0.0125 − 0.00408i)7-s + (1.20 + 0.392i)8-s + (4.20 + 3.05i)9-s + 0.356i·10-s + (6.77 − 8.66i)11-s − 7.75·12-s + (3.39 − 4.67i)13-s + (0.000651 − 0.00200i)14-s + (−1.34 − 4.14i)15-s + (−12.6 + 9.22i)16-s + (−17.8 − 24.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0468 − 0.0644i)2-s + (−0.200 + 0.618i)3-s + (0.307 + 0.945i)4-s + (−0.361 + 0.262i)5-s + (0.0304 + 0.0419i)6-s + (0.00179 − 0.000583i)7-s + (0.151 + 0.0491i)8-s + (0.466 + 0.339i)9-s + 0.0356i·10-s + (0.616 − 0.787i)11-s − 0.646·12-s + (0.261 − 0.359i)13-s + (4.65e−5 − 0.000143i)14-s + (−0.0898 − 0.276i)15-s + (−0.793 + 0.576i)16-s + (−1.04 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.974991 + 0.632830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974991 + 0.632830i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.80 - 1.31i)T \) |
| 11 | \( 1 + (-6.77 + 8.66i)T \) |
| good | 2 | \( 1 + (-0.0936 + 0.128i)T + (-1.23 - 3.80i)T^{2} \) |
| 3 | \( 1 + (0.602 - 1.85i)T + (-7.28 - 5.29i)T^{2} \) |
| 7 | \( 1 + (-0.0125 + 0.00408i)T + (39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-3.39 + 4.67i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (17.8 + 24.5i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-12.4 - 4.03i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 - 13.0T + 529T^{2} \) |
| 29 | \( 1 + (-27.2 + 8.86i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-10.2 - 7.46i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (8.14 + 25.0i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-2.87 - 0.933i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 73.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-6.67 + 20.5i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (55.8 + 40.6i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (18.9 + 58.2i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (56.5 + 77.8i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 67.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (41.3 - 30.0i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (41.1 - 13.3i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (4.49 - 6.18i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (0.721 + 0.992i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 117.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-106. - 77.6i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−15.85443441187677052845677914486, −14.09542559467638864798423703347, −12.97732716157304661703658042083, −11.60134920690568579360118636308, −10.96170324815448010576794840919, −9.371510093758658077962256888102, −7.991165107587407898105706674005, −6.72388322695164616008034185032, −4.64480615497035529434112458894, −3.18818771425762984152907227252,
1.49639468571285230802312960616, 4.45516816485103377154957278940, 6.26000434721326809175091486122, 7.15008973866106776981374707150, 8.947514828715297823902983132753, 10.24955509158682261816095222362, 11.52932520611932127849014546233, 12.54198766888066049034633809416, 13.72517997110996518955571941978, 15.05168994297607470498430042473
