| L(s) = 1 | + 4.87·2-s + 15.7·4-s + 7·7-s + 37.6·8-s − 36.9·11-s − 61.3·13-s + 34.1·14-s + 57.7·16-s + 44.8·17-s − 139.·19-s − 180.·22-s − 217.·23-s − 298.·26-s + 110.·28-s + 33.8·29-s + 124.·31-s − 20.2·32-s + 218.·34-s + 237.·37-s − 680.·38-s − 195.·41-s − 343.·43-s − 582.·44-s − 1.06e3·46-s − 16.8·47-s + 49·49-s − 964.·52-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.96·4-s + 0.377·7-s + 1.66·8-s − 1.01·11-s − 1.30·13-s + 0.651·14-s + 0.902·16-s + 0.639·17-s − 1.68·19-s − 1.74·22-s − 1.97·23-s − 2.25·26-s + 0.743·28-s + 0.216·29-s + 0.720·31-s − 0.111·32-s + 1.10·34-s + 1.05·37-s − 2.90·38-s − 0.743·41-s − 1.21·43-s − 1.99·44-s − 3.39·46-s − 0.0522·47-s + 0.142·49-s − 2.57·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.87T + 8T^{2} \) |
| 11 | \( 1 + 36.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 135.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 100.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 880.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 619.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 231.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.260192996195327555376963987093, −7.77638572977771886411205310562, −6.71008137028231825129289898588, −6.02907360431688516182933584501, −5.12210358817457716828225045792, −4.60207766339489869732154192801, −3.71995528740725014078178535884, −2.58183003656032716491662161671, −2.01805424338462969312087899668, 0,
2.01805424338462969312087899668, 2.58183003656032716491662161671, 3.71995528740725014078178535884, 4.60207766339489869732154192801, 5.12210358817457716828225045792, 6.02907360431688516182933584501, 6.71008137028231825129289898588, 7.77638572977771886411205310562, 8.260192996195327555376963987093
