| L(s) = 1 | − 2.85i·2-s − 12.3·3-s + 23.8·4-s − 38.2i·5-s + 35.0i·6-s − 96.3·7-s − 159. i·8-s − 91.5·9-s − 109.·10-s − 564.·11-s − 293.·12-s − 678. i·13-s + 274. i·14-s + 471. i·15-s + 309.·16-s + 2.07e3i·17-s + ⋯ |
| L(s) = 1 | − 0.504i·2-s − 0.789·3-s + 0.745·4-s − 0.684i·5-s + 0.398i·6-s − 0.742·7-s − 0.880i·8-s − 0.376·9-s − 0.345·10-s − 1.40·11-s − 0.588·12-s − 1.11i·13-s + 0.374i·14-s + 0.540i·15-s + 0.302·16-s + 1.74i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.134159 - 0.780806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134159 - 0.780806i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-7.84e3 + 2.77e3i)T \) |
| good | 2 | \( 1 + 2.85iT - 32T^{2} \) |
| 3 | \( 1 + 12.3T + 243T^{2} \) |
| 5 | \( 1 + 38.2iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 96.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 564.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 678. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.07e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.69e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.07e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.96e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.19e3iT - 2.86e7T^{2} \) |
| 41 | \( 1 + 6.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.58e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.36e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.44e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.98e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.64e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.31e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.65e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.35e5iT - 8.58e9T^{2} \) |
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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
−15.20055843605482396208778089653, −12.84919515847155178507892056686, −12.64028538089749619934455187383, −10.96922128245656566266156990572, −10.35614246906741889482081569019, −8.415222410426053524481575788321, −6.59076500078821840263710410064, −5.25806415352943953938201291114, −2.86544094791093219607220650424, −0.46372924119616669686295576353,
2.76291891723758200801295723885, 5.48262770753398428979497077744, 6.55989999147115385934107380879, 7.73182521158877369602000045335, 9.880868716842671361883038522680, 11.15925243372079521088372338901, 11.94129096020543758468919883815, 13.66927609756915510966495561213, 14.92979663894558984412668456947, 16.20890925948863655445891978666
