| L(s) = 1 | + (0.540 + 0.841i)2-s + (−1.66 − 0.239i)3-s + (−0.415 + 0.909i)4-s + (−1.39 − 1.74i)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 + (0.715 − 2.11i)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 + (1.90 + 3.24i)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 + (−0.624 − 0.781i)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 + (0.226 − 0.669i)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 + (0.492 + 0.838i)15-s + (−0.163 − 0.188i)16-s + (−0.582 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0213686 - 0.0626408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0213686 - 0.0626408i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.39 + 1.74i)T \) |
| 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
−12.12326096785218773813013938775, −11.19329423154415065146558289457, −9.799527396883833483174545495480, −8.736775377697865989843838837018, −7.68576110410409985832624388362, −6.56076560112357925618669871857, −5.48607351969769838622922785780, −4.83936597452398696623715110488, −3.13562246660006388292221319821, −0.05067183201070653306382366305,
2.69582686761624989354899681381, 4.08921750624903804762886486976, 5.10855072209680840708375051615, 6.62398398432463432111990172556, 7.10994886391714401539828104806, 8.939304665497198591972461062409, 10.20361734945395030666160153404, 10.71271667095648397891603229788, 11.77475695911302076510996672816, 12.12008049450480916980220558592
