| L(s) = 1 | + 4.41i·5-s − 3.24·7-s − 0.171i·11-s − 14.4·25-s − 2.82i·29-s − 9.24·31-s − 14.3i·35-s + 3.51·49-s − 4.07i·53-s + 0.757·55-s + 11.3i·59-s − 15.4·73-s + 0.556i·77-s + 10·79-s + 17.8i·83-s + ⋯ |
| L(s) = 1 | + 1.97i·5-s − 1.22·7-s − 0.0517i·11-s − 2.89·25-s − 0.525i·29-s − 1.66·31-s − 2.41i·35-s + 0.502·49-s − 0.559i·53-s + 0.102·55-s + 1.47i·59-s − 1.81·73-s + 0.0634i·77-s + 1.12·79-s + 1.95i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.602548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.602548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.41iT - 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 0.171iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.07iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−10.52177129329211553704031242622, −9.911433569964495430821275752986, −9.137025727202356671305563826299, −7.76454176126931624460895685317, −7.03769540968738111850309825999, −6.41643064417248935499572875009, −5.66263481165500652538892055716, −3.89893133420175609926305559963, −3.20411459514129297215561107937, −2.30862903333473820109330197806,
0.27759676442585793268376754337, 1.71039540932488136867696777869, 3.38068711076203999920707988420, 4.38143671440090286964158824815, 5.31412908788555003760873862525, 6.07180940167385785464572069016, 7.24149046544847600839073552901, 8.227104100961930092838766765804, 9.110379666334423814965364212116, 9.422085931366405110389971791871
