L(s) = 1 | − 0.590·2-s − 3-s − 1.65·4-s + 0.503·5-s + 0.590·6-s − 2.98·7-s + 2.15·8-s + 9-s − 0.297·10-s − 0.330·11-s + 1.65·12-s − 13-s + 1.76·14-s − 0.503·15-s + 2.03·16-s + 6.92·17-s − 0.590·18-s − 0.460·19-s − 0.832·20-s + 2.98·21-s + 0.195·22-s − 8.61·23-s − 2.15·24-s − 4.74·25-s + 0.590·26-s − 27-s + 4.93·28-s + ⋯ |
L(s) = 1 | − 0.417·2-s − 0.577·3-s − 0.825·4-s + 0.225·5-s + 0.240·6-s − 1.12·7-s + 0.761·8-s + 0.333·9-s − 0.0940·10-s − 0.0996·11-s + 0.476·12-s − 0.277·13-s + 0.470·14-s − 0.130·15-s + 0.508·16-s + 1.68·17-s − 0.139·18-s − 0.105·19-s − 0.186·20-s + 0.651·21-s + 0.0415·22-s − 1.79·23-s − 0.439·24-s − 0.949·25-s + 0.115·26-s − 0.192·27-s + 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.590T + 2T^{2} \) |
| 5 | \( 1 - 0.503T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 + 0.330T + 11T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 0.460T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 - 0.0773T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 6.05T + 97T^{2} \) |
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
−7.997212363715043097475040526050, −7.52312344732388816264803548784, −6.54574095024043233217705716973, −5.71312885855708583183083599254, −5.36359848144793129992054896979, −4.10876424597222643952922410067, −3.66642265984506925278087676461, −2.38149965366863975175352914447, −1.04307123592928699463913032495, 0,
1.04307123592928699463913032495, 2.38149965366863975175352914447, 3.66642265984506925278087676461, 4.10876424597222643952922410067, 5.36359848144793129992054896979, 5.71312885855708583183083599254, 6.54574095024043233217705716973, 7.52312344732388816264803548784, 7.997212363715043097475040526050