| L(s) = 1 | + 448. i·3-s + 1.39e3·5-s + 1.93e4i·7-s − 1.42e5·9-s + 2.07e5i·11-s − 6.51e4·13-s + 6.27e5i·15-s − 8.05e5·17-s − 4.08e6i·19-s − 8.67e6·21-s − 4.40e6i·23-s + 1.95e6·25-s − 3.73e7i·27-s + 1.06e7·29-s − 2.90e6i·31-s + ⋯ |
| L(s) = 1 | + 1.84i·3-s + 0.447·5-s + 1.15i·7-s − 2.40·9-s + 1.28i·11-s − 0.175·13-s + 0.825i·15-s − 0.567·17-s − 1.64i·19-s − 2.12·21-s − 0.684i·23-s + 0.200·25-s − 2.60i·27-s + 0.516·29-s − 0.101i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.6862759975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6862759975\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.39e3T \) |
| good | 3 | \( 1 - 448. iT - 5.90e4T^{2} \) |
| 7 | \( 1 - 1.93e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.07e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 6.51e4T + 1.37e11T^{2} \) |
| 17 | \( 1 + 8.05e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 4.08e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 4.40e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 1.06e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 2.90e6iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 1.45e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 6.38e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.40e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.51e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 3.71e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 4.69e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 7.97e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 8.39e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 4.87e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.05e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 4.58e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 2.95e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 3.76e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 1.37e10T + 7.37e19T^{2} \) |
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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
−9.868148993200021132963198349018, −9.089108418785313316116270179262, −8.637663020839218753461626799054, −6.91890647494734186543675261802, −5.70802414184279342652528310315, −4.89168943310483045703700678181, −4.30197738905668292939955100140, −2.85828002134695258854986998487, −2.22300467776751254268279685893, −0.13348902100802976203104470064,
0.928355771822517616264232928144, 1.47482522226577443634894640534, 2.65525912740919313029270824501, 3.75871982614252810448236052509, 5.49883971800254053074613310203, 6.28671555974256496251380751280, 7.04553383186986041062872449939, 7.944054387978673362791224173932, 8.571584639088707030189336212735, 9.987760646182407611721714213508
