| L(s) = 1 | − 1.48i·2-s − 0.806i·3-s − 0.193·4-s − 1.19·6-s + 3.35i·7-s − 2.67i·8-s + 2.35·9-s + 0.962·11-s + 0.156i·12-s − 6.15i·13-s + 4.96·14-s − 4.35·16-s − 6.31i·17-s − 3.48i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 1.04i·2-s − 0.465i·3-s − 0.0969·4-s − 0.487·6-s + 1.26i·7-s − 0.945i·8-s + 0.783·9-s + 0.290·11-s + 0.0451i·12-s − 1.70i·13-s + 1.32·14-s − 1.08·16-s − 1.53i·17-s − 0.820i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.860484 - 1.39229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860484 - 1.39229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48iT - 2T^{2} \) |
| 3 | \( 1 + 0.806iT - 3T^{2} \) |
| 7 | \( 1 - 3.35iT - 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 + 6.15iT - 13T^{2} \) |
| 17 | \( 1 + 6.31iT - 17T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 4.12iT - 43T^{2} \) |
| 47 | \( 1 - 3.35iT - 47T^{2} \) |
| 53 | \( 1 + 1.84iT - 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.73iT - 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 + 0.387iT - 73T^{2} \) |
| 79 | \( 1 - 0.836T + 79T^{2} \) |
| 83 | \( 1 - 7.03iT - 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−10.87330483658798523731711821941, −9.840112875962938741536600039919, −9.304425900635043248687814289381, −7.979247868708084433214581644409, −7.11773524871055099062767365417, −5.99240968010511142263142228909, −4.91071574355800502009757620631, −3.30449075657326734063347055999, −2.49977397895818048729073091438, −1.13577466378416316519171220224,
1.79980934376102832186888373327, 3.97165143462199310648785119715, 4.45394485049696642444098347691, 5.91032526669856359768868508896, 6.91051756611928927108608369606, 7.28939368428451597780305742248, 8.491244452330510267636194237337, 9.374241071299322481098771766690, 10.52087605087791118043096793097, 10.96793069604540662808276854664
