| L(s) = 1 | + (0.545 + 0.491i)2-s + (−0.994 + 0.104i)3-s + (−0.152 − 1.45i)4-s + (−2.19 + 0.430i)5-s + (−0.593 − 0.431i)6-s + (1.23 + 2.34i)7-s + (1.49 − 2.05i)8-s + (0.978 − 0.207i)9-s + (−1.40 − 0.843i)10-s + (−2.07 − 0.440i)11-s + (0.303 + 1.42i)12-s + (−4.39 − 1.42i)13-s + (−0.476 + 1.88i)14-s + (2.13 − 0.657i)15-s + (−1.03 + 0.219i)16-s + (−2.19 − 4.92i)17-s + ⋯ |
| L(s) = 1 | + (0.385 + 0.347i)2-s + (−0.574 + 0.0603i)3-s + (−0.0763 − 0.726i)4-s + (−0.981 + 0.192i)5-s + (−0.242 − 0.176i)6-s + (0.466 + 0.884i)7-s + (0.527 − 0.726i)8-s + (0.326 − 0.0693i)9-s + (−0.445 − 0.266i)10-s + (−0.624 − 0.132i)11-s + (0.0876 + 0.412i)12-s + (−1.21 − 0.395i)13-s + (−0.127 + 0.503i)14-s + (0.551 − 0.169i)15-s + (−0.258 + 0.0549i)16-s + (−0.531 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0409894 - 0.177984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0409894 - 0.177984i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (2.19 - 0.430i)T \) |
| 7 | \( 1 + (-1.23 - 2.34i)T \) |
| good | 2 | \( 1 + (-0.545 - 0.491i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (2.07 + 0.440i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (4.39 + 1.42i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.19 + 4.92i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.687 - 6.54i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (6.23 + 5.61i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (2.19 - 1.59i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (9.10 - 4.05i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.0106 - 0.0501i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (0.0644 - 0.198i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.35iT - 43T^{2} \) |
| 47 | \( 1 + (-2.52 + 5.66i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-3.67 + 0.386i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-4.24 - 4.71i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-9.37 + 10.4i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-5.55 - 12.4i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-10.0 + 7.33i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.335 - 1.57i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (4.42 + 1.97i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.26 - 3.11i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.72 - 3.02i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (7.72 + 10.6i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.53963253979727864332460830031, −9.816485808727672417397689386636, −8.580358896283386534921676229630, −7.59349341087336349424381201491, −6.76642499569309733445777181595, −5.53367591182408408252910495001, −5.07989719593015310897446394347, −3.98187847053066529890657952096, −2.26512933296250125122432597578, −0.095453764837352895696764137457,
2.16964914509671565524357960781, 3.81889578030270390616605709969, 4.38540914861795834995034055514, 5.29974854595565047770028235050, 7.02925941554130897176569902559, 7.55667336654687451014179010980, 8.296021970257783759862840996390, 9.568286278914603702519651281883, 10.84829862171737724168077123935, 11.26109031953333268381420371427
