| L(s) = 1 | + (−0.721 + 0.143i)3-s + (−2.80 + 4.19i)5-s + (−0.203 − 0.304i)7-s + (−7.81 + 3.23i)9-s + (−1.28 + 6.47i)11-s + (−7.51 + 7.51i)13-s + (1.42 − 3.43i)15-s + (−15.2 + 7.43i)17-s + (22.0 + 9.12i)19-s + (0.190 + 0.190i)21-s + (−4.56 − 0.908i)23-s + (−0.181 − 0.438i)25-s + (10.6 − 7.13i)27-s + (−0.758 − 0.506i)29-s + (−7.88 − 39.6i)31-s + ⋯ |
| L(s) = 1 | + (−0.240 + 0.0478i)3-s + (−0.560 + 0.839i)5-s + (−0.0290 − 0.0435i)7-s + (−0.868 + 0.359i)9-s + (−0.117 + 0.588i)11-s + (−0.577 + 0.577i)13-s + (0.0947 − 0.228i)15-s + (−0.899 + 0.437i)17-s + (1.15 + 0.480i)19-s + (0.00908 + 0.00908i)21-s + (−0.198 − 0.0394i)23-s + (−0.00726 − 0.0175i)25-s + (0.395 − 0.264i)27-s + (−0.0261 − 0.0174i)29-s + (−0.254 − 1.27i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.320557 + 0.654044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320557 + 0.654044i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (15.2 - 7.43i)T \) |
| good | 3 | \( 1 + (0.721 - 0.143i)T + (8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (2.80 - 4.19i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (0.203 + 0.304i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (1.28 - 6.47i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (7.51 - 7.51i)T - 169iT^{2} \) |
| 19 | \( 1 + (-22.0 - 9.12i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (4.56 + 0.908i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (0.758 + 0.506i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (7.88 + 39.6i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (5.52 - 1.09i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-14.2 - 21.2i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-57.7 + 23.9i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (15.6 - 15.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-51.0 - 21.1i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-15.7 - 38.0i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (33.6 - 22.4i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 11.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (80.3 - 15.9i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (44.5 - 66.6i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (17.2 - 86.9i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (12.2 - 29.6i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-92.6 - 92.6i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (7.48 + 5.00i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−13.41554827378799407857282193949, −12.05010056658604366230471610253, −11.35793870855370409242520159878, −10.43817362105140485268663284910, −9.237937210665556605138025926370, −7.85213859031812138022884051866, −6.95710330936571688424848567893, −5.61627765340877703745269807730, −4.12140466005101863345910986073, −2.53670323968195120426327759326,
0.48716022496140677060712236017, 3.05188273560327719924824244709, 4.73738729685449561692020294819, 5.78443632963273369783153318219, 7.29204746992203267763316757465, 8.495077108284374958123304473685, 9.258958064810144504718647976083, 10.76426615287747490502135069378, 11.73531456089855898560366999225, 12.43776045471900082515266489867
