| L(s) = 1 | + (0.387 + 1.89i)2-s + (0.987 + 0.160i)3-s + (−1.60 + 0.684i)4-s + (−3.18 + 0.785i)5-s + (0.0779 + 1.93i)6-s + (−0.0361 + 0.0761i)7-s + (0.278 + 0.402i)8-s + (0.948 + 0.316i)9-s + (−2.72 − 5.74i)10-s + (−1.87 + 0.626i)11-s + (−1.69 + 0.418i)12-s + (0.194 + 3.60i)13-s + (−0.158 − 0.0390i)14-s + (−3.27 + 0.264i)15-s + (−3.07 + 3.20i)16-s + (−2.58 + 5.43i)17-s + ⋯ |
| L(s) = 1 | + (0.273 + 1.34i)2-s + (0.569 + 0.0926i)3-s + (−0.803 + 0.342i)4-s + (−1.42 + 0.351i)5-s + (0.0318 + 0.789i)6-s + (−0.0136 + 0.0287i)7-s + (0.0983 + 0.142i)8-s + (0.316 + 0.105i)9-s + (−0.861 − 1.81i)10-s + (−0.565 + 0.188i)11-s + (−0.489 + 0.120i)12-s + (0.0540 + 0.998i)13-s + (−0.0423 − 0.0104i)14-s + (−0.845 + 0.0682i)15-s + (−0.769 + 0.800i)16-s + (−0.625 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144673 - 1.24113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144673 - 1.24113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (-0.194 - 3.60i)T \) |
| good | 2 | \( 1 + (-0.387 - 1.89i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (3.18 - 0.785i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (0.0361 - 0.0761i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (1.87 - 0.626i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (2.58 - 5.43i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (3.08 + 5.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.72 + 4.71i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.19 - 5.85i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (4.20 + 2.20i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-8.03 + 5.07i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (1.67 + 0.272i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-7.42 - 4.69i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-0.807 - 6.64i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-6.56 - 9.51i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-1.83 - 1.91i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-8.54 - 0.689i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (3.72 + 1.58i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (0.0492 - 0.0603i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (2.12 + 1.88i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (1.30 + 10.7i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (2.02 - 5.35i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (3.02 - 5.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.82 - 9.75i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−11.06060805705160768797267006917, −10.84363035655115445754182943889, −9.023157899571770719459742674914, −8.525250384577689859309391546247, −7.55129676044967336714102851459, −7.05435198570264440739157551596, −6.10829902834592201175779951320, −4.53915800020928778943725989914, −4.14569824699154561174317207381, −2.51444003837807101312598209591,
0.63117619573867668232097422738, 2.40948511273780090169767468883, 3.44796191278037849691369208382, 4.14337535840392839341891329303, 5.27715389037489588999262275317, 7.11437197948706308413589606246, 7.87207316810524406019260861646, 8.674744436366980430944222236130, 9.804409951033623199555649894736, 10.59524689805228634252890715902
