| L(s) = 1 | + 7.17i·3-s + 146.·7-s + 191.·9-s − 42.4i·11-s − 605. i·13-s − 409.·17-s + 2.09e3i·19-s + 1.05e3i·21-s + 3.04e3·23-s + 3.11e3i·27-s − 4.59e3i·29-s + 5.11e3·31-s + 304.·33-s + 1.12e4i·37-s + 4.34e3·39-s + ⋯ |
| L(s) = 1 | + 0.460i·3-s + 1.13·7-s + 0.787·9-s − 0.105i·11-s − 0.993i·13-s − 0.343·17-s + 1.33i·19-s + 0.521i·21-s + 1.20·23-s + 0.823i·27-s − 1.01i·29-s + 0.956·31-s + 0.0486·33-s + 1.35i·37-s + 0.457·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.086906260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.086906260\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 7.17iT - 243T^{2} \) |
| 7 | \( 1 - 146.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 42.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 605. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 409.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.09e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.41e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.82e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.66e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.18e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.18e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.65e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.59e4T + 8.58e9T^{2} \) |
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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
−9.768496002939734944563483097928, −8.539490279692281147706043076854, −7.984611430478173661300989958657, −7.07705476611249373661141429009, −5.89764563336920450322581160381, −4.96483143381181934248603353897, −4.26757542057922451393832469678, −3.15858909260490306365077297133, −1.82314086989951402339424844007, −0.828848983944286625660381769607,
0.884455768236903648558753741234, 1.70558177722114426431134418563, 2.74899494984655170796537347294, 4.39810530459990310320841920481, 4.73453192846653348493136701915, 6.09085387593076073819713014009, 7.16869886095243824160524667465, 7.46558115749947867459370786891, 8.779294751824258253009848482988, 9.209785822604717020529461402008
