| 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
−10.24875642006252447952241018384, −9.313060015961307458530779984313, −8.796635924682996660342435748979, −7.75465425815915971386712795979, −6.78177424502527471420094530003, −5.99450815449592522591125190951, −4.13358198422322150890978338145, −3.10334985191545992703641541796, −1.23438658473595018547536285964, 0,
1.23438658473595018547536285964, 3.10334985191545992703641541796, 4.13358198422322150890978338145, 5.99450815449592522591125190951, 6.78177424502527471420094530003, 7.75465425815915971386712795979, 8.796635924682996660342435748979, 9.313060015961307458530779984313, 10.24875642006252447952241018384
