| L(s) = 1 | + 6.32·3-s + 25·5-s + 44.2·7-s − 203·9-s − 720.·11-s − 146·13-s + 158.·15-s − 702·17-s + 2.73e3·19-s + 280·21-s − 4.09e3·23-s + 625·25-s − 2.82e3·27-s − 4.01e3·29-s + 4.56e3·31-s − 4.56e3·33-s + 1.10e3·35-s − 1.47e4·37-s − 923.·39-s − 4.35e3·41-s + 1.24e4·43-s − 5.07e3·45-s + 6.01e3·47-s − 1.48e4·49-s − 4.43e3·51-s − 1.81e4·53-s − 1.80e4·55-s + ⋯ |
| L(s) = 1 | + 0.405·3-s + 0.447·5-s + 0.341·7-s − 0.835·9-s − 1.79·11-s − 0.239·13-s + 0.181·15-s − 0.589·17-s + 1.73·19-s + 0.138·21-s − 1.61·23-s + 0.200·25-s − 0.744·27-s − 0.885·29-s + 0.853·31-s − 0.728·33-s + 0.152·35-s − 1.77·37-s − 0.0972·39-s − 0.404·41-s + 1.02·43-s − 0.373·45-s + 0.397·47-s − 0.883·49-s − 0.239·51-s − 0.887·53-s − 0.803·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 6.32T + 243T^{2} \) |
| 7 | \( 1 - 44.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 720.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 146T + 3.71e5T^{2} \) |
| 17 | \( 1 + 702T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.01e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.67e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.68e4T + 8.58e9T^{2} \) |
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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
−11.44079852339033313904747732595, −10.41559393037844674671989022670, −9.456921831135260988493960799406, −8.259752739285223596484178124134, −7.52267883304160237339431920722, −5.85417839002029544098591972680, −4.99702212153681142991005519872, −3.16595762683206554473585709810, −2.07183874084935133448829511495, 0,
2.07183874084935133448829511495, 3.16595762683206554473585709810, 4.99702212153681142991005519872, 5.85417839002029544098591972680, 7.52267883304160237339431920722, 8.259752739285223596484178124134, 9.456921831135260988493960799406, 10.41559393037844674671989022670, 11.44079852339033313904747732595
