| L(s) = 1 | + (0.739 + 0.673i)2-s + (−0.932 − 0.361i)3-s + (0.0922 + 0.995i)4-s + (−2.35 + 1.45i)5-s + (−0.445 − 0.895i)6-s + (−0.0712 − 0.0943i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−2.72 − 0.508i)10-s + (−2.76 − 2.51i)11-s + (0.273 − 0.961i)12-s + (−3.60 − 4.77i)13-s + (0.0109 − 0.117i)14-s + (2.72 − 0.508i)15-s + (−0.982 + 0.183i)16-s + (1.82 − 3.65i)17-s + ⋯ |
| L(s) = 1 | + (0.522 + 0.476i)2-s + (−0.538 − 0.208i)3-s + (0.0461 + 0.497i)4-s + (−1.05 + 0.651i)5-s + (−0.181 − 0.365i)6-s + (−0.0269 − 0.0356i)7-s + (−0.213 + 0.282i)8-s + (0.246 + 0.224i)9-s + (−0.860 − 0.160i)10-s + (−0.833 − 0.759i)11-s + (0.0789 − 0.277i)12-s + (−1.00 − 1.32i)13-s + (0.00291 − 0.0314i)14-s + (0.702 − 0.131i)15-s + (−0.245 + 0.0459i)16-s + (0.441 − 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.266637 - 0.332598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266637 - 0.332598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.739 - 0.673i)T \) |
| 3 | \( 1 + (0.932 + 0.361i)T \) |
| 103 | \( 1 + (-2.27 + 9.89i)T \) |
| good | 5 | \( 1 + (2.35 - 1.45i)T + (2.22 - 4.47i)T^{2} \) |
| 7 | \( 1 + (0.0712 + 0.0943i)T + (-1.91 + 6.73i)T^{2} \) |
| 11 | \( 1 + (2.76 + 2.51i)T + (1.01 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.60 + 4.77i)T + (-3.55 + 12.5i)T^{2} \) |
| 17 | \( 1 + (-1.82 + 3.65i)T + (-10.2 - 13.5i)T^{2} \) |
| 19 | \( 1 + (-7.50 + 2.90i)T + (14.0 - 12.8i)T^{2} \) |
| 23 | \( 1 + (4.45 - 4.06i)T + (2.12 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.537 + 0.332i)T + (12.9 - 25.9i)T^{2} \) |
| 31 | \( 1 + (7.70 - 1.44i)T + (28.9 - 11.1i)T^{2} \) |
| 37 | \( 1 + (-0.728 + 2.55i)T + (-31.4 - 19.4i)T^{2} \) |
| 41 | \( 1 + (1.97 + 1.22i)T + (18.2 + 36.7i)T^{2} \) |
| 43 | \( 1 + (2.45 + 8.62i)T + (-36.5 + 22.6i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (4.73 - 1.83i)T + (39.1 - 35.7i)T^{2} \) |
| 59 | \( 1 + (0.504 - 0.668i)T + (-16.1 - 56.7i)T^{2} \) |
| 61 | \( 1 + (1.43 - 2.88i)T + (-36.7 - 48.6i)T^{2} \) |
| 67 | \( 1 + (5.26 - 6.96i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-4.85 - 3.00i)T + (31.6 + 63.5i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 6.95i)T + (32.5 + 65.3i)T^{2} \) |
| 79 | \( 1 + (8.97 - 5.55i)T + (35.2 - 70.7i)T^{2} \) |
| 83 | \( 1 + (4.42 + 5.86i)T + (-22.7 + 79.8i)T^{2} \) |
| 89 | \( 1 + (0.476 - 5.13i)T + (-87.4 - 16.3i)T^{2} \) |
| 97 | \( 1 + (-4.85 - 9.74i)T + (-58.4 + 77.4i)T^{2} \) |
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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
−10.57134229443052584332339020303, −9.630343047589793874358392087570, −8.116688528010602818880582283885, −7.49974821499638274465965082839, −7.05417244901421081248499478219, −5.45778437359656549155298300437, −5.25609623638141889945735640175, −3.59512808845092536842833393432, −2.88457149903294531883265846463, −0.20551404361752094919602886351,
1.75217073987188910735748800934, 3.42444828142103805239068282621, 4.49899422910054756911480085589, 4.97315786146568661943874361984, 6.17580506114023070010320750990, 7.40123008100382283704792029076, 8.087258930847695946532578528445, 9.501208005528432676620686232639, 10.00011346713764438994003188926, 11.10768807135507231119054368572
