| L(s) = 1 | + (1.15 − 0.0550i)2-s + (1.68 + 0.419i)3-s + (−0.658 + 0.0628i)4-s + (1.35 + 1.29i)5-s + (1.96 + 0.392i)6-s + (−0.328 − 1.70i)7-s + (−3.04 + 0.438i)8-s + (2.64 + 1.40i)9-s + (1.64 + 1.42i)10-s + (1.28 − 0.662i)11-s + (−1.13 − 0.170i)12-s + (−3.21 − 0.619i)13-s + (−0.473 − 1.95i)14-s + (1.74 + 2.74i)15-s + (−2.20 + 0.424i)16-s + (−0.539 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (0.817 − 0.0389i)2-s + (0.970 + 0.242i)3-s + (−0.329 + 0.0314i)4-s + (0.608 + 0.579i)5-s + (0.802 + 0.160i)6-s + (−0.124 − 0.644i)7-s + (−1.07 + 0.154i)8-s + (0.882 + 0.469i)9-s + (0.519 + 0.450i)10-s + (0.387 − 0.199i)11-s + (−0.326 − 0.0491i)12-s + (−0.891 − 0.171i)13-s + (−0.126 − 0.521i)14-s + (0.449 + 0.709i)15-s + (−0.550 + 0.106i)16-s + (−0.130 − 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08990 + 0.298089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08990 + 0.298089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.68 - 0.419i)T \) |
| 23 | \( 1 + (4.56 - 1.46i)T \) |
| good | 2 | \( 1 + (-1.15 + 0.0550i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (-1.35 - 1.29i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.328 + 1.70i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-1.28 + 0.662i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (3.21 + 0.619i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (0.539 + 1.18i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.474 - 0.216i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.911 + 9.54i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (4.12 - 3.24i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (1.84 - 6.27i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.29 + 3.45i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (6.23 - 7.92i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (0.800 + 0.462i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.01 - 6.94i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.52 + 7.88i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (1.99 - 4.97i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-0.990 + 1.92i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-2.73 - 4.26i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.08 - 4.57i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (12.2 - 4.23i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-1.97 + 1.87i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (0.261 - 1.81i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 2.71i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.79154179154010863747829147551, −11.67841287128547267511343931125, −10.14523756535116761603091355543, −9.708920718787281998970820896355, −8.515012140577766443794123173615, −7.33666427265599973265412989181, −6.10404220919026061576471895882, −4.68959680551967891248419178620, −3.67760319612830349353708234277, −2.50494481551351236930013470052,
2.10338271175987192519557051285, 3.58459808030923072446469716821, 4.82155326793806220690136614970, 5.87950495544707321630599115752, 7.19387837859194880499105989128, 8.704912625690133577427172823795, 9.164035324028065576307199796266, 10.06685403191443897412087333008, 11.97976338892303892849762054160, 12.64268540740112170957137829012
