| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.0512 + 1.73i)3-s + (−0.959 − 0.281i)4-s + (0.182 − 2.22i)5-s + (−1.72 − 0.195i)6-s + (1.96 + 1.26i)7-s + (0.415 − 0.909i)8-s + (−2.99 + 0.177i)9-s + (2.17 + 0.497i)10-s + (0.590 + 4.10i)11-s + (0.438 − 1.67i)12-s + (−2.60 − 4.05i)13-s + (−1.53 + 1.76i)14-s + (3.86 + 0.201i)15-s + (0.841 + 0.540i)16-s + (2.07 + 7.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.0295 + 0.999i)3-s + (−0.479 − 0.140i)4-s + (0.0815 − 0.996i)5-s + (−0.702 − 0.0798i)6-s + (0.743 + 0.477i)7-s + (0.146 − 0.321i)8-s + (−0.998 + 0.0590i)9-s + (0.689 + 0.157i)10-s + (0.178 + 1.23i)11-s + (0.126 − 0.483i)12-s + (−0.722 − 1.12i)13-s + (−0.409 + 0.472i)14-s + (0.998 + 0.0520i)15-s + (0.210 + 0.135i)16-s + (0.503 + 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.311442 + 1.18382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.311442 + 1.18382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.0512 - 1.73i)T \) |
| 5 | \( 1 + (-0.182 + 2.22i)T \) |
| 23 | \( 1 + (-0.585 - 4.75i)T \) |
| good | 7 | \( 1 + (-1.96 - 1.26i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.590 - 4.10i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.60 + 4.05i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.07 - 7.06i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.43 - 4.89i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.892 - 3.03i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.520 + 1.14i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.29 + 1.49i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.22 - 7.12i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.57 - 5.64i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + (-7.10 + 11.0i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.99 - 3.10i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (5.51 + 2.51i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.160 - 1.11i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.381 + 0.0548i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.69 + 15.9i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (3.72 + 5.79i)T + (-32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (12.0 + 10.4i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.45 - 14.1i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.68 - 6.56i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.36582226291803879289530633873, −9.960538178831377020209233466186, −9.037744673933171029935785405038, −8.189128214758353356179718914220, −7.74971647645418273212605002020, −6.04815050713553592590566287115, −5.31312793199705573241280500918, −4.68327048346688749311007353100, −3.67714484859845323432720871738, −1.76269836321314135681241538812,
0.67989649743329645784800154727, 2.26478345132948109036012924614, 2.99294790671465679869667435390, 4.41763962653462065360750939494, 5.64425210151439090342563421118, 6.91906973859898049498123570863, 7.27223562453324418092578372275, 8.445359131695165964968641171897, 9.191135133100317061329433002308, 10.34910715506483704439912196518
