| L(s) = 1 | + 243·9-s + 1.19e3i·13-s + 2.24e3i·17-s − 2.95e3·29-s − 1.22e4i·37-s − 2.09e4·41-s + 1.68e4·49-s − 7.29e3i·53-s + 1.89e4·61-s + 8.88e4i·73-s + 5.90e4·81-s − 5.10e4·89-s − 9.21e4i·97-s − 9.80e4·101-s − 2.46e5·109-s + ⋯ |
| L(s) = 1 | + 9-s + 1.95i·13-s + 1.88i·17-s − 0.651·29-s − 1.47i·37-s − 1.94·41-s + 49-s − 0.356i·53-s + 0.652·61-s + 1.95i·73-s + 0.999·81-s − 0.683·89-s − 0.994i·97-s − 0.955·101-s − 1.98·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.270753557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270753557\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 243T^{2} \) |
| 7 | \( 1 - 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.19e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.24e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.47e6T^{2} \) |
| 23 | \( 1 - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.29e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.88e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.21e4iT - 8.58e9T^{2} \) |
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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
−9.865183485112014052122778136501, −9.052533305039025467714829190533, −8.252411624352257470782930037893, −7.11706389193744869845964303185, −6.59301053333020968344039660404, −5.50010840423586224670129878272, −4.22288725083518325994894470244, −3.83295438662945671772894708822, −2.07123365537866385116726341342, −1.44123920693509882309558987390,
0.25741620326941965962957600079, 1.23391464300200731199668140643, 2.65278835601502608275604455844, 3.52350676840174299477922582326, 4.82027764880071655049357762733, 5.42917153698859629405464819500, 6.67952714958749669696416758267, 7.45836490995697305659893686496, 8.166694900410464455209788023336, 9.282887394621082613278032723394
