| L(s) = 1 | + (10.1 + 11.8i)3-s + 19.3i·5-s − 82.7i·7-s + (−38.1 + 239. i)9-s + 332.·11-s − 1.17e3·13-s + (−229. + 195. i)15-s − 1.98e3i·17-s − 1.34e3i·19-s + (981. − 837. i)21-s + 1.83e3·23-s + 2.75e3·25-s + (−3.23e3 + 1.97e3i)27-s + 1.92e3i·29-s − 5.80e3i·31-s + ⋯ |
| L(s) = 1 | + (0.649 + 0.760i)3-s + 0.345i·5-s − 0.638i·7-s + (−0.157 + 0.987i)9-s + 0.828·11-s − 1.92·13-s + (−0.262 + 0.224i)15-s − 1.66i·17-s − 0.855i·19-s + (0.485 − 0.414i)21-s + 0.722·23-s + 0.880·25-s + (−0.853 + 0.521i)27-s + 0.424i·29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.922818467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922818467\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.1 - 11.8i)T \) |
| good | 5 | \( 1 - 19.3iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 82.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 332.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.34e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.92e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.80e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 6.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.15e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 8.37e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.05e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 9.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.79e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 4.83e4T + 8.58e9T^{2} \) |
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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
−10.22240222372461917173682895461, −9.500643905784451648137921839930, −8.826350622172224293743208764043, −7.30327690107185931235398260364, −7.07713758850523692950030145464, −5.15463380990266108402313666293, −4.50483787712383050606678417278, −3.22141994391909141254619593505, −2.33823397786665091291901307363, −0.44333883428693760051494763568,
1.23290262095998803697305439788, 2.23758831344732717934923059654, 3.42102490701856147253878715166, 4.76049025514239998884491281525, 6.02019124710968042730812404292, 6.94466198055851245993181242501, 7.926978777768779415987823838341, 8.775920103272496828880931286725, 9.469365890327408249387281218210, 10.54927665393148108266830864320
