| L(s) = 1 | + (−1.41 − 1.27i)2-s + (−0.994 + 0.104i)3-s + (0.172 + 1.63i)4-s + (−2.15 − 0.585i)5-s + (1.54 + 1.12i)6-s + (2.41 − 1.07i)7-s + (−0.394 + 0.542i)8-s + (0.978 − 0.207i)9-s + (2.31 + 3.58i)10-s + (3.83 + 0.815i)11-s + (−0.342 − 1.61i)12-s + (1.04 + 0.339i)13-s + (−4.80 − 1.56i)14-s + (2.20 + 0.357i)15-s + (4.47 − 0.952i)16-s + (0.815 + 1.83i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 − 0.903i)2-s + (−0.574 + 0.0603i)3-s + (0.0861 + 0.819i)4-s + (−0.965 − 0.261i)5-s + (0.630 + 0.458i)6-s + (0.913 − 0.406i)7-s + (−0.139 + 0.191i)8-s + (0.326 − 0.0693i)9-s + (0.731 + 1.13i)10-s + (1.15 + 0.245i)11-s + (−0.0989 − 0.465i)12-s + (0.290 + 0.0942i)13-s + (−1.28 − 0.417i)14-s + (0.569 + 0.0921i)15-s + (1.11 − 0.238i)16-s + (0.197 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.300498 - 0.524753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.300498 - 0.524753i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.15 + 0.585i)T \) |
| 7 | \( 1 + (-2.41 + 1.07i)T \) |
| good | 2 | \( 1 + (1.41 + 1.27i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (-3.83 - 0.815i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 0.339i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.815 - 1.83i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.0467 - 0.444i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (6.01 + 5.41i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.29 + 2.39i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.88 + 0.840i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (1.72 + 8.10i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-0.178 + 0.549i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 5.19iT - 43T^{2} \) |
| 47 | \( 1 + (-1.48 + 3.32i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (2.11 - 0.222i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.77 + 1.96i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (7.57 - 8.41i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 3.07i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 9.07i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.02 + 4.81i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (1.56 + 0.696i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.60 - 9.08i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.49 - 1.65i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-1.09 - 1.50i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.71172916167901207090501204960, −9.878973866785391118154955168254, −8.765582694421591660837495184386, −8.207160295428159079410234592088, −7.22363756337855516419219016737, −5.96085660929830023099479850809, −4.55255402676969031606640959829, −3.76821880712243286639038050632, −1.88095547969314711399079766551, −0.66981273148106957121904180736,
1.19006918131496412528667168669, 3.47911447140387239507658714637, 4.70439871940314985171967002996, 5.98609475137077598430606001148, 6.75574127198094894844988795069, 7.70654356100437093065079873987, 8.271594815581702362670989753654, 9.143220198881414360115732776173, 10.11916139606384883650007352437, 11.24594536990740297622203006214
