| L(s) = 1 | + 24.2i·5-s + 58i·7-s − 13.8·11-s + 20.7i·13-s + 306·17-s + 602.·19-s + 468i·23-s + 37·25-s + 1.46e3i·29-s + 110i·31-s − 1.40e3·35-s − 1.03e3i·37-s − 2.97e3·41-s + 2.88e3·43-s − 396i·47-s + ⋯ |
| L(s) = 1 | + 0.969i·5-s + 1.18i·7-s − 0.114·11-s + 0.122i·13-s + 1.05·17-s + 1.66·19-s + 0.884i·23-s + 0.0592·25-s + 1.74i·29-s + 0.114i·31-s − 1.14·35-s − 0.759i·37-s − 1.76·41-s + 1.56·43-s − 0.179i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.929336023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929336023\) |
| \(L(3)\) |
|
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 | \( 1 - 24.2iT - 625T^{2} \) |
| 7 | \( 1 - 58iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 13.8T + 1.46e4T^{2} \) |
| 13 | \( 1 - 20.7iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 306T + 8.35e4T^{2} \) |
| 19 | \( 1 - 602.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 468iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.46e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 110iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.03e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.97e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.88e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 396iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.12e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.68e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.98e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.80e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.58e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.48e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 13.8T + 4.74e7T^{2} \) |
| 89 | \( 1 + 8.76e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.91e3T + 8.85e7T^{2} \) |
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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
−10.45085953295990005611144793697, −9.571696307597721673304371860127, −8.812714951223009955799413263895, −7.64609791055940916240710073499, −6.96853020118629211389195352374, −5.75912214044975777687463040411, −5.19108827014840352275216159500, −3.43941429242788714875623886050, −2.81874914104307685120001305152, −1.43059986297073727279643692032,
0.53589296286443654262950760966, 1.30226530012618004123728893309, 3.05382735755918681139550269290, 4.19854130125694695183558640409, 5.02888363041396077347805217448, 6.05334778555968845348108606937, 7.34578781610736621137443006633, 7.900591899914201678465857322466, 8.937703627283658690198028953553, 9.905198413074510719724238491748
