| L(s) = 1 | + (0.576 − 0.0504i)2-s + (−1.63 + 0.289i)4-s + (−2.18 + 0.474i)5-s + (2.78 − 1.94i)7-s + (−2.05 + 0.549i)8-s + (−1.23 + 0.384i)10-s + (−1.55 − 4.27i)11-s + (0.357 − 4.08i)13-s + (1.50 − 1.26i)14-s + (1.97 − 0.718i)16-s + (−0.284 − 0.0763i)17-s + (−4.26 − 2.46i)19-s + (3.44 − 1.40i)20-s + (−1.11 − 2.38i)22-s + (−0.869 + 1.24i)23-s + ⋯ |
| L(s) = 1 | + (0.407 − 0.0356i)2-s + (−0.819 + 0.144i)4-s + (−0.977 + 0.212i)5-s + (1.05 − 0.735i)7-s + (−0.724 + 0.194i)8-s + (−0.391 + 0.121i)10-s + (−0.469 − 1.29i)11-s + (0.0990 − 1.13i)13-s + (0.402 − 0.337i)14-s + (0.493 − 0.179i)16-s + (−0.0690 − 0.0185i)17-s + (−0.977 − 0.564i)19-s + (0.770 − 0.315i)20-s + (−0.237 − 0.509i)22-s + (−0.181 + 0.258i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516143 - 0.686986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516143 - 0.686986i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.18 - 0.474i)T \) |
| good | 2 | \( 1 + (-0.576 + 0.0504i)T + (1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-2.78 + 1.94i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.55 + 4.27i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.357 + 4.08i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (0.284 + 0.0763i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (4.26 + 2.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.869 - 1.24i)T + (-7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (5.57 + 4.67i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.591 - 3.35i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.28 + 4.81i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.42 - 2.89i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.02 - 8.63i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.09 + 1.56i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 1.39i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.34 - 3.40i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.249 - 1.41i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.46 + 0.827i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (0.728 - 0.420i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.166 - 0.619i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.56 + 11.3i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.830 + 9.49i)T + (-81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.19 + 10.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.11 + 1.91i)T + (62.3 + 74.3i)T^{2} \) |
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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
−11.06448304433624141964499707873, −10.37995545032286037013954211139, −8.873555960513611827885911156982, −8.115428188394728243237833343694, −7.60827499088081837247808888577, −5.99456217092375016645097995072, −4.91343197785101461156684593812, −4.04421690645868050168307367964, −3.07104463240617239207274799478, −0.50920318828066062126285147583,
1.98898667802344902535989582487, 3.91434284833110129896782238954, 4.61771288614927823543125072563, 5.40608137928698047481770703733, 6.86097460541650331172415745031, 8.023613729890716632663163495604, 8.671064049446031755622657978471, 9.550329011031555622450251192695, 10.74846795350537868928299929509, 11.78435968353359007532249450503
