| L(s) = 1 | + (−0.761 − 0.366i)2-s + (−0.802 − 1.00i)4-s + (1.87 − 1.73i)5-s + (3.43 + 1.98i)7-s + (0.617 + 2.70i)8-s + (−2.06 + 0.636i)10-s + (−2.73 − 2.17i)11-s + (4.94 + 1.52i)13-s + (−1.88 − 2.76i)14-s + (−0.0507 + 0.222i)16-s + (1.07 − 1.15i)17-s + (−1.13 − 0.444i)19-s + (−3.25 − 0.490i)20-s + (1.28 + 2.65i)22-s + (0.829 − 5.50i)23-s + ⋯ |
| L(s) = 1 | + (−0.538 − 0.259i)2-s + (−0.401 − 0.502i)4-s + (0.838 − 0.777i)5-s + (1.29 + 0.749i)7-s + (0.218 + 0.956i)8-s + (−0.652 + 0.201i)10-s + (−0.823 − 0.656i)11-s + (1.37 + 0.422i)13-s + (−0.504 − 0.739i)14-s + (−0.0126 + 0.0555i)16-s + (0.259 − 0.280i)17-s + (−0.259 − 0.101i)19-s + (−0.727 − 0.109i)20-s + (0.272 + 0.566i)22-s + (0.172 − 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04320 - 0.611890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04320 - 0.611890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (0.707 + 6.51i)T \) |
| good | 2 | \( 1 + (0.761 + 0.366i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-1.87 + 1.73i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (-3.43 - 1.98i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.73 + 2.17i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.94 - 1.52i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 1.15i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.13 + 0.444i)T + (13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (-0.829 + 5.50i)T + (-21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (2.39 - 1.63i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.484 + 6.46i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (-2.81 + 1.62i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.35 - 6.95i)T + (-25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-5.52 + 4.40i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-2.63 - 8.55i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-3.33 - 0.760i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (11.4 + 0.860i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (5.34 - 13.6i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (10.0 - 1.51i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (0.861 - 2.79i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-6.64 + 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.08 - 3.05i)T + (-30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (1.44 + 0.985i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (12.1 - 15.2i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−11.01666412544018508471594738732, −10.31229549850674483840791597921, −9.004089346864422333170692710936, −8.784085300272973636796187503495, −7.916673332627170172536341861745, −5.96303325510136896896691746601, −5.43801564350271818024361324568, −4.48498665423471746689884520746, −2.28826937468891119953522297619, −1.18956174799678135947407054486,
1.58107531021821797067470284663, 3.33426422456113968128284519698, 4.56574464406377000088157838478, 5.79497052812336407964826603301, 7.06594571078020576547464425401, 7.80625962299374239295358197668, 8.540917155038968657057846722651, 9.729068833656191321125757281746, 10.52568899031284760117302689011, 11.09269454136774139804596434929
