| L(s) = 1 | + (−1.02 + 0.592i)2-s + (−1.29 + 2.24i)4-s − 4.21i·5-s + (−4.97 + 8.62i)7-s − 7.81i·8-s + (2.49 + 4.32i)10-s + (−16.4 + 9.48i)11-s + (−4.13 − 12.3i)13-s − 11.7i·14-s + (−0.568 − 0.985i)16-s + (−7.09 − 4.09i)17-s + (−11.0 + 19.1i)19-s + (9.48 + 5.47i)20-s + (11.2 − 19.4i)22-s + (−17.6 + 10.1i)23-s + ⋯ |
| L(s) = 1 | + (−0.512 + 0.296i)2-s + (−0.324 + 0.562i)4-s − 0.843i·5-s + (−0.711 + 1.23i)7-s − 0.976i·8-s + (0.249 + 0.432i)10-s + (−1.49 + 0.862i)11-s + (−0.317 − 0.948i)13-s − 0.842i·14-s + (−0.0355 − 0.0615i)16-s + (−0.417 − 0.241i)17-s + (−0.582 + 1.00i)19-s + (0.474 + 0.273i)20-s + (0.510 − 0.884i)22-s + (−0.765 + 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0236i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00322433 + 0.272381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00322433 + 0.272381i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (4.13 + 12.3i)T \) |
| good | 2 | \( 1 + (1.02 - 0.592i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + 4.21iT - 25T^{2} \) |
| 7 | \( 1 + (4.97 - 8.62i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (16.4 - 9.48i)T + (60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (7.09 + 4.09i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (11.0 - 19.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (17.6 - 10.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-9.66 + 5.57i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 11.3T + 961T^{2} \) |
| 37 | \( 1 + (-25.2 - 43.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.23 + 5.33i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-30.1 + 52.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 - 71.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 12.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (43.4 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (31.6 - 54.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (36.7 + 63.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (84.7 + 48.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 - 9.79T + 5.32e3T^{2} \) |
| 79 | \( 1 + 17.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (143. - 83.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (53.9 - 93.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.43604911743916067964168482849, −12.48209122367487680196326069182, −12.37219072659551869683126469865, −10.27433509618899227676565235623, −9.416609328141303647486785419469, −8.402596863685913262899236293884, −7.63880632841028694239039277822, −5.93416614306719412719490583954, −4.67555486449492630533226253282, −2.76338745102619428089748818268,
0.21646368705124327691493199509, 2.64966639719665695037307089638, 4.45353632572984827610588953786, 6.13560181458890394945539965163, 7.25082428153658241872637847467, 8.605003574319413291860313141310, 9.883044009218133080443989390253, 10.63652151737667072518578390215, 11.16578591122558418034938377567, 13.03680440870399575162926685041
