| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 2·7-s − 1.73·8-s + 11-s + 1.46·13-s − 3.46·14-s − 5·16-s − 1.46·19-s + 1.73·22-s − 6.92·23-s + 2.53·26-s − 1.99·28-s − 3.46·29-s + 2.92·31-s − 5.19·32-s − 8.92·37-s − 2.53·38-s + 3.46·41-s − 8.92·43-s + 0.999·44-s − 11.9·46-s + 6.92·47-s − 3·49-s + 1.46·52-s − 12.9·53-s + 3.46·56-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.755·7-s − 0.612·8-s + 0.301·11-s + 0.406·13-s − 0.925·14-s − 1.25·16-s − 0.335·19-s + 0.369·22-s − 1.44·23-s + 0.497·26-s − 0.377·28-s − 0.643·29-s + 0.525·31-s − 0.918·32-s − 1.46·37-s − 0.411·38-s + 0.541·41-s − 1.36·43-s + 0.150·44-s − 1.76·46-s + 1.01·47-s − 0.428·49-s + 0.203·52-s − 1.77·53-s + 0.462·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−8.607522049627000704940417341322, −7.66149334077370576666142539244, −6.54973818543039877891372524390, −6.21411664904222640178475119308, −5.37114300791228383583758529819, −4.46344441284911895297876364495, −3.71454119464702276352815475694, −3.07618491095699059708805535565, −1.88100372135999436373448821297, 0,
1.88100372135999436373448821297, 3.07618491095699059708805535565, 3.71454119464702276352815475694, 4.46344441284911895297876364495, 5.37114300791228383583758529819, 6.21411664904222640178475119308, 6.54973818543039877891372524390, 7.66149334077370576666142539244, 8.607522049627000704940417341322
