| L(s) = 1 | + 3.74·2-s + 6.05·4-s + 5·5-s − 31.3·7-s − 7.30·8-s + 18.7·10-s − 20.8·11-s + 59.9·13-s − 117.·14-s − 75.7·16-s − 74.0·17-s − 63.8·19-s + 30.2·20-s − 78.0·22-s − 32.8·23-s + 25·25-s + 224.·26-s − 189.·28-s − 160.·29-s − 254.·31-s − 225.·32-s − 277.·34-s − 156.·35-s + 215.·37-s − 239.·38-s − 36.5·40-s + 141.·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.756·4-s + 0.447·5-s − 1.69·7-s − 0.322·8-s + 0.592·10-s − 0.571·11-s + 1.27·13-s − 2.24·14-s − 1.18·16-s − 1.05·17-s − 0.770·19-s + 0.338·20-s − 0.756·22-s − 0.297·23-s + 0.200·25-s + 1.69·26-s − 1.28·28-s − 1.02·29-s − 1.47·31-s − 1.24·32-s − 1.40·34-s − 0.756·35-s + 0.956·37-s − 1.02·38-s − 0.144·40-s + 0.539·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 3.74T + 8T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 + 20.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 141.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 41.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 615.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 134.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 857.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 588.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 618.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 414.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 201.T + 9.12e5T^{2} \) |
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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
−10.57644366265537770157240056742, −9.427229710000842326459043743265, −8.766238457940642009707002295898, −7.08981863602694559404947191349, −6.14154487269679994398155530644, −5.74661470241413914151714312229, −4.25785469816814921017030590132, −3.43674415870720298916800428285, −2.35049074009932263747777905854, 0,
2.35049074009932263747777905854, 3.43674415870720298916800428285, 4.25785469816814921017030590132, 5.74661470241413914151714312229, 6.14154487269679994398155530644, 7.08981863602694559404947191349, 8.766238457940642009707002295898, 9.427229710000842326459043743265, 10.57644366265537770157240056742
