| L(s) = 1 | + (−1.09 − 0.744i)3-s + (−1.30 + 1.21i)5-s + (0.00749 − 0.0129i)7-s + (−0.458 − 1.16i)9-s + (1.29 − 1.62i)11-s + (−1.55 − 0.478i)13-s + (2.32 − 0.350i)15-s + (4.42 + 4.10i)17-s + (−2.78 + 7.09i)19-s + (−0.0178 + 0.00858i)21-s + (−5.24 − 0.790i)23-s + (−0.136 + 1.82i)25-s + (−1.25 + 5.47i)27-s + (−5.92 + 4.04i)29-s + (−0.178 − 2.37i)31-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.429i)3-s + (−0.583 + 0.541i)5-s + (0.00283 − 0.00490i)7-s + (−0.152 − 0.389i)9-s + (0.390 − 0.490i)11-s + (−0.430 − 0.132i)13-s + (0.600 − 0.0905i)15-s + (1.07 + 0.995i)17-s + (−0.638 + 1.62i)19-s + (−0.00389 + 0.00187i)21-s + (−1.09 − 0.164i)23-s + (−0.0273 + 0.364i)25-s + (−0.240 + 1.05i)27-s + (−1.10 + 0.750i)29-s + (−0.0320 − 0.427i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287123 + 0.410910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287123 + 0.410910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-1.84 - 6.29i)T \) |
| good | 3 | \( 1 + (1.09 + 0.744i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (1.30 - 1.21i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (-0.00749 + 0.0129i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 1.62i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.55 + 0.478i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-4.42 - 4.10i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.78 - 7.09i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (5.24 + 0.790i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (5.92 - 4.04i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.178 + 2.37i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (2.52 + 4.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.20 + 1.54i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-6.24 - 7.83i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (6.55 - 2.02i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.370 - 1.62i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.452 - 6.03i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (0.523 - 1.33i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (6.96 - 1.05i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-9.32 - 2.87i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-6.00 + 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.49 + 5.10i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (13.1 + 8.95i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-8.44 + 10.5i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.83990639043466900615477726405, −10.07114824737987688361728779540, −8.973034237273257006620442761496, −7.901481053237624484645789601170, −7.31179146164518223914302144611, −6.01864255400365252515393145660, −5.82478636322993036712495311411, −4.05178330901847654818968554103, −3.32525558135378258389558571845, −1.53265245374146038517813596851,
0.28965460130047561351176008239, 2.28909726707280505796569714006, 3.87670958951893646511107827202, 4.79597345725484392106121606506, 5.40391830182401671100949246857, 6.68902596032520467396632473652, 7.60357434010658030316314707656, 8.477166729147318930014630659271, 9.489337689225559904992252235168, 10.18288923583377869708187835857
