| L(s) = 1 | + 3.09i·3-s + 25i·5-s − 118.·7-s + 233.·9-s − 355. i·11-s + 370. i·13-s − 77.4·15-s − 586.·17-s − 972. i·19-s − 368. i·21-s − 1.29e3·23-s − 625·25-s + 1.47e3i·27-s − 5.22e3i·29-s + 4.62e3·31-s + ⋯ |
| L(s) = 1 | + 0.198i·3-s + 0.447i·5-s − 0.916·7-s + 0.960·9-s − 0.885i·11-s + 0.608i·13-s − 0.0889·15-s − 0.491·17-s − 0.618i·19-s − 0.182i·21-s − 0.511·23-s − 0.200·25-s + 0.389i·27-s − 1.15i·29-s + 0.863·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.761205589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761205589\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
| good | 3 | \( 1 - 3.09iT - 243T^{2} \) |
| 7 | \( 1 + 118.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 355. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 370. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 586.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 972. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.22e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.29e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.40e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.22e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.09e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.85e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.62e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.68e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.33e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.83e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−10.70009048923387246043141471822, −9.810927038965696382543059355950, −9.097967836505092665449418687026, −7.83558713077779779473515016120, −6.73316897681717089379969941756, −6.09649015696152634775259455087, −4.54383189029991764012126219036, −3.56027531838422138889092644959, −2.33648850538947142255851319127, −0.61828648213851562301151332799,
0.880576596402362640833090949994, 2.22065827508355485272717699526, 3.71548742700760195809364508674, 4.74305815341734912714125809555, 6.02602519023910124375550979513, 7.01103442219142082184390951132, 7.86706330074559424475143139030, 9.108027245873665877135790368674, 9.903198539375278802917274114944, 10.61178065538704774461028810252
