| L(s) = 1 | + (−0.540 + 0.841i)2-s + (1.66 − 0.239i)3-s + (−0.415 − 0.909i)4-s + (−0.229 − 2.22i)5-s + (−0.699 + 1.53i)6-s + (1.93 + 1.67i)7-s + (0.989 + 0.142i)8-s + (−0.155 + 0.0456i)9-s + (1.99 + 1.00i)10-s + (−0.0282 + 0.0181i)11-s + (−0.910 − 1.41i)12-s + (5.06 − 4.38i)13-s + (−2.45 + 0.722i)14-s + (−0.915 − 3.65i)15-s + (−0.654 + 0.755i)16-s + (2.40 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (0.962 − 0.138i)3-s + (−0.207 − 0.454i)4-s + (−0.102 − 0.994i)5-s + (−0.285 + 0.625i)6-s + (0.732 + 0.634i)7-s + (0.349 + 0.0503i)8-s + (−0.0517 + 0.0152i)9-s + (0.630 + 0.319i)10-s + (−0.00853 + 0.00548i)11-s + (−0.262 − 0.409i)12-s + (1.40 − 1.21i)13-s + (−0.657 + 0.192i)14-s + (−0.236 − 0.943i)15-s + (−0.163 + 0.188i)16-s + (0.582 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38839 + 0.0827703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38839 + 0.0827703i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (0.229 + 2.22i)T \) |
| 23 | \( 1 + (4.09 - 2.49i)T \) |
| good | 3 | \( 1 + (-1.66 + 0.239i)T + (2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.93 - 1.67i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.0282 - 0.0181i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-5.06 + 4.38i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 1.09i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.0156 - 0.0342i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.504 + 1.10i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.815 - 5.67i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.14 - 3.91i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (10.1 + 2.98i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.37 + 0.197i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 4.78iT - 47T^{2} \) |
| 53 | \( 1 + (7.67 + 6.64i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (2.32 + 2.68i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.93 - 13.4i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-6.64 + 10.3i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (6.16 + 3.96i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.28 - 1.95i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.36 - 7.35i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.38 - 8.11i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.613 - 4.26i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.15 + 14.1i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−12.36641184051757104647317849816, −11.24710342052927644712054473083, −9.978285134356648892539704176080, −8.799475870064950973419995345284, −8.339581780612121753486334198705, −7.79249074879151876555683644726, −5.95951492212697373396914520506, −5.13035948635865393379817114401, −3.48135588829753877373794750618, −1.58240121479891051909584764313,
1.93187455854690372142695771151, 3.33991735169433591823627185252, 4.19279500980969044937212536258, 6.26254683560046612523106149972, 7.54570456293031810268856786050, 8.317017044705681863406379579710, 9.280942106010014432888151355773, 10.30336188085941711378453284060, 11.19830662824474896801790004028, 11.78568660946428391168732817528
