| L(s) = 1 | + 3.90·2-s + 3·3-s + 7.25·4-s + 7.35·5-s + 11.7·6-s − 2.90·8-s + 9·9-s + 28.7·10-s − 11·11-s + 21.7·12-s − 60.6·13-s + 22.0·15-s − 69.3·16-s + 4.18·17-s + 35.1·18-s − 88.2·19-s + 53.4·20-s − 42.9·22-s − 73.9·23-s − 8.70·24-s − 70.8·25-s − 236.·26-s + 27·27-s − 232.·29-s + 86.2·30-s − 101.·31-s − 247.·32-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.907·4-s + 0.658·5-s + 0.797·6-s − 0.128·8-s + 0.333·9-s + 0.908·10-s − 0.301·11-s + 0.523·12-s − 1.29·13-s + 0.379·15-s − 1.08·16-s + 0.0597·17-s + 0.460·18-s − 1.06·19-s + 0.597·20-s − 0.416·22-s − 0.670·23-s − 0.0740·24-s − 0.566·25-s − 1.78·26-s + 0.192·27-s − 1.48·29-s + 0.524·30-s − 0.587·31-s − 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.90T + 8T^{2} \) |
| 5 | \( 1 - 7.35T + 125T^{2} \) |
| 13 | \( 1 + 60.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.18T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 530.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 496.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 716.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 907.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 79.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 862.T + 9.12e5T^{2} \) |
| 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
−8.740697856269295392154586412677, −7.58800994067944384630014612710, −6.93061467473152519000263348212, −5.83021231811386784400263917965, −5.38527596994738790920904132130, −4.33329345804775163663888814215, −3.69687179166572071506141796359, −2.47330779572100003938819724000, −2.05554342746935608488702754912, 0,
2.05554342746935608488702754912, 2.47330779572100003938819724000, 3.69687179166572071506141796359, 4.33329345804775163663888814215, 5.38527596994738790920904132130, 5.83021231811386784400263917965, 6.93061467473152519000263348212, 7.58800994067944384630014612710, 8.740697856269295392154586412677
