| L(s) = 1 | + (−0.00801 − 0.00721i)2-s + (0.994 − 0.104i)3-s + (−0.209 − 1.98i)4-s + (1.81 + 1.30i)5-s + (−0.00872 − 0.00633i)6-s + (2.56 + 0.665i)7-s + (−0.0253 + 0.0349i)8-s + (0.978 − 0.207i)9-s + (−0.00516 − 0.0235i)10-s + (−3.19 − 0.679i)11-s + (−0.415 − 1.95i)12-s + (5.64 + 1.83i)13-s + (−0.0157 − 0.0238i)14-s + (1.94 + 1.10i)15-s + (−3.91 + 0.831i)16-s + (−1.27 − 2.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.00566 − 0.00510i)2-s + (0.574 − 0.0603i)3-s + (−0.104 − 0.994i)4-s + (0.812 + 0.582i)5-s + (−0.00356 − 0.00258i)6-s + (0.967 + 0.251i)7-s + (−0.00896 + 0.0123i)8-s + (0.326 − 0.0693i)9-s + (−0.00163 − 0.00744i)10-s + (−0.964 − 0.205i)11-s + (−0.120 − 0.564i)12-s + (1.56 + 0.508i)13-s + (−0.00420 − 0.00636i)14-s + (0.501 + 0.285i)15-s + (−0.977 + 0.207i)16-s + (−0.309 − 0.695i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04804 - 0.358539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04804 - 0.358539i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.81 - 1.30i)T \) |
| 7 | \( 1 + (-2.56 - 0.665i)T \) |
| good | 2 | \( 1 + (0.00801 + 0.00721i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (3.19 + 0.679i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.64 - 1.83i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.27 + 2.86i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.407 - 3.87i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.69 + 2.42i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.956 - 0.695i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.462 + 0.205i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.09 + 9.87i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-3.38 + 10.4i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.65iT - 43T^{2} \) |
| 47 | \( 1 + (2.65 - 5.96i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (5.28 - 0.555i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.0160 - 0.0178i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-5.08 + 5.64i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.822 - 1.84i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (7.85 - 5.70i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.52 - 11.9i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (5.81 + 2.58i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (5.80 - 7.98i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.42 - 4.91i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (11.0 + 15.2i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.84831024091822996700535610333, −9.967466165049324290383085360831, −9.048948196851951905034176829427, −8.321251700410176012929958175775, −7.14307904746361917947061549290, −6.01505397677980025079744685559, −5.41998655787276902955803403328, −4.10053782956838076655003681554, −2.48121628065763208254239968706, −1.55686761197785073323653288457,
1.64870282972654021441233943211, 2.94292065680370598384507116551, 4.19503487728065360832905554817, 5.08761568776452682083268428956, 6.34244564849922117619348355005, 7.66691169531032155196892752756, 8.353431695105480304930865800691, 8.782704447674458194657553552422, 9.987336151312960852802007032704, 10.87649848312055943993531681547
