| L(s) = 1 | − 1.01e3i·5-s − 907.·7-s + 9.54e3i·11-s + 1.99e3·13-s − 4.61e4i·17-s − 3.41e4·19-s + 9.70e4i·23-s − 6.37e5·25-s + 8.33e5i·29-s − 1.03e6·31-s + 9.20e5i·35-s − 1.96e6·37-s + 1.24e6i·41-s + 1.48e6·43-s − 1.69e6i·47-s + ⋯ |
| L(s) = 1 | − 1.62i·5-s − 0.377·7-s + 0.651i·11-s + 0.0699·13-s − 0.552i·17-s − 0.261·19-s + 0.346i·23-s − 1.63·25-s + 1.17i·29-s − 1.11·31-s + 0.613i·35-s − 1.05·37-s + 0.442i·41-s + 0.434·43-s − 0.348i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.025957073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025957073\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 907.T \) |
| good | 5 | \( 1 + 1.01e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 9.54e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.99e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.61e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 3.41e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 9.70e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.33e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.03e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.96e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.24e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.48e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.69e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.91e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.43e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.12e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.96e5T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.02e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.22e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.61e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.08e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.60e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.04e7T + 7.83e15T^{2} \) |
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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
−10.71930493570051594258715295195, −9.500377638646406195491800818965, −9.006187895188297111959820964020, −7.956344245520651405877975836060, −6.85424275759931693711831831880, −5.46824277548086944377249775405, −4.76847738441489504227173232651, −3.61116769803685060069496701016, −1.96875491658786882858499698085, −0.905263087362227383086597661377,
0.25768752592726922572654975847, 2.05237630152015289788238391085, 3.09800828352467005112704342212, 3.95256435079105340148386258587, 5.71224525606945727910075975631, 6.51053633868327684758680287198, 7.35637805916385425770322214716, 8.471929301684257746151201134224, 9.702717955743173810425190776279, 10.61996983090401723199083059130
