| L(s) = 1 | + (−2 − 7.74i)2-s + (−56.0 + 30.9i)4-s − 10·5-s + 309. i·7-s + (352. + 371. i)8-s + (20 + 77.4i)10-s + 960. i·11-s + 1.46e3·13-s + (2.40e3 − 619. i)14-s + (2.17e3 − 3.47e3i)16-s + 4.76e3·17-s + 7.52e3i·19-s + (560. − 309. i)20-s + (7.44e3 − 1.92e3i)22-s + 1.04e4i·23-s + ⋯ |
| L(s) = 1 | + (−0.250 − 0.968i)2-s + (−0.875 + 0.484i)4-s − 0.0800·5-s + 0.903i·7-s + (0.687 + 0.726i)8-s + (0.0200 + 0.0774i)10-s + 0.721i·11-s + 0.667·13-s + (0.874 − 0.225i)14-s + (0.531 − 0.847i)16-s + 0.970·17-s + 1.09i·19-s + (0.0700 − 0.0387i)20-s + (0.698 − 0.180i)22-s + 0.860i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.08316 + 0.279672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08316 + 0.279672i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 + 7.74i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 10T + 1.56e4T^{2} \) |
| 7 | \( 1 - 309. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 960. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.46e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.76e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.52e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.04e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.54e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.18e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.99e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.93e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 2.15e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.56e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.92e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 7.84e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.09e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.94e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 5.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.88e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.10e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.04e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.10e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.45e6T + 8.32e11T^{2} \) |
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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
−15.19755129769794762188397093459, −13.86133369304127629027848202170, −12.50234514752396569369057100319, −11.74401993254377110396440551467, −10.28644249148582617274419606406, −9.146381875245853150472495329173, −7.82144820074768550819296707183, −5.49372218812668412870995172440, −3.58534030091463156957275559967, −1.73440716424836894128099930297,
0.66216327723945447924290492526, 3.99108376444755018349914647240, 5.77345908724164017939585401655, 7.21204729706262020207796872116, 8.421464492921012040452393287853, 9.854141353182004647642123314349, 11.16483195714808959246272943063, 13.16477310494905267473887220700, 13.97532729971438139979261159079, 15.20556111485963827446348057610
