| L(s) = 1 | − 2-s − 2.40·3-s + 4-s − 2.70·5-s + 2.40·6-s + 0.173·7-s − 8-s + 2.77·9-s + 2.70·10-s − 2.75·11-s − 2.40·12-s + 4.76·13-s − 0.173·14-s + 6.50·15-s + 16-s − 5.93·17-s − 2.77·18-s − 19-s − 2.70·20-s − 0.416·21-s + 2.75·22-s + 4.10·23-s + 2.40·24-s + 2.33·25-s − 4.76·26-s + 0.548·27-s + 0.173·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.21·5-s + 0.980·6-s + 0.0654·7-s − 0.353·8-s + 0.923·9-s + 0.856·10-s − 0.831·11-s − 0.693·12-s + 1.32·13-s − 0.0462·14-s + 1.67·15-s + 0.250·16-s − 1.44·17-s − 0.653·18-s − 0.229·19-s − 0.605·20-s − 0.0907·21-s + 0.588·22-s + 0.855·23-s + 0.490·24-s + 0.466·25-s − 0.933·26-s + 0.105·27-s + 0.0327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 0.173T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 0.785T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 + 0.994T + 47T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 8.01T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 - 9.30T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
| 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
−8.555185356683982946899277053691, −8.107811456459977680640612465274, −7.11403438231860480739264090853, −6.50555129353105286298235584613, −5.71279714901598232137506705760, −4.72922076318216684425144907881, −3.95402139038303092425826228599, −2.66034596372910487082817222423, −1.01902680210026560204573482019, 0,
1.01902680210026560204573482019, 2.66034596372910487082817222423, 3.95402139038303092425826228599, 4.72922076318216684425144907881, 5.71279714901598232137506705760, 6.50555129353105286298235584613, 7.11403438231860480739264090853, 8.107811456459977680640612465274, 8.555185356683982946899277053691
