| L(s) = 1 | + 6.32i·3-s + 17.8i·5-s − 22.6·7-s − 13.0·9-s − 44.2i·11-s − 17.8i·13-s − 113.·15-s + 70·17-s + 82.2i·19-s − 143. i·21-s − 158.·23-s − 195.·25-s + 88.5i·27-s + 125. i·29-s + 280·33-s + ⋯ |
| L(s) = 1 | + 1.21i·3-s + 1.59i·5-s − 1.22·7-s − 0.481·9-s − 1.21i·11-s − 0.381i·13-s − 1.94·15-s + 0.998·17-s + 0.992i·19-s − 1.48i·21-s − 1.43·23-s − 1.56·25-s + 0.631i·27-s + 0.801i·29-s + 1.47·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.03176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03176i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.32iT - 27T^{2} \) |
| 5 | \( 1 - 17.8iT - 125T^{2} \) |
| 7 | \( 1 + 22.6T + 343T^{2} \) |
| 11 | \( 1 + 44.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 17.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 375. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 182T + 6.89e4T^{2} \) |
| 43 | \( 1 - 132. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 82.2iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 232. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 221. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 910T + 3.89e5T^{2} \) |
| 79 | \( 1 - 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 714. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 546T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490T + 9.12e5T^{2} \) |
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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
−13.67899429290759603254175071055, −12.19068465020443144312628544609, −10.93403934088402298490618874132, −10.20748072813757285961104199978, −9.694926080346705034738807088031, −8.082688529071269249181523519771, −6.61216925073114069213018639094, −5.70107624936722437144644396843, −3.62644832231848533663856288426, −3.15706713652802246004115415058,
0.52240111283131630807705731875, 2.01542653512916390805880529757, 4.22586096781788846847025218758, 5.71317941436571232451105364763, 6.93904359422186065305981708579, 7.912089215880290909214454444923, 9.179098137008116716528552416196, 9.947518204547707664068174667346, 12.03581223370086710500124360580, 12.40832890459561949129560088840
