| L(s) = 1 | − 0.381·2-s − 1.85·4-s + 1.47·8-s + 3.14·16-s − 8.23·17-s + 4.70·19-s − 7.47·23-s − 10.7·31-s − 4.14·32-s + 3.14·34-s − 1.79·38-s + 2.85·46-s − 8.94·47-s − 7·49-s − 9.76·53-s + 12.4·61-s + 4.09·62-s − 4.70·64-s + 15.2·68-s − 8.72·76-s − 1.29·79-s + 5.94·83-s + 13.8·92-s + 3.41·94-s + 2.67·98-s + 3.72·106-s + 17.8·107-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.927·4-s + 0.520·8-s + 0.786·16-s − 1.99·17-s + 1.08·19-s − 1.55·23-s − 1.92·31-s − 0.732·32-s + 0.539·34-s − 0.291·38-s + 0.420·46-s − 1.30·47-s − 49-s − 1.34·53-s + 1.58·61-s + 0.519·62-s − 0.588·64-s + 1.85·68-s − 1.00·76-s − 0.145·79-s + 0.652·83-s + 1.44·92-s + 0.352·94-s + 0.270·98-s + 0.362·106-s + 1.72·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 8.23T + 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 9.76T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 1.29T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
| 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
−9.846806652314932081194942554342, −9.262817905128557648662036611928, −8.430775478361451180134409531171, −7.62146689440114469921833710882, −6.53745925572367705019854657846, −5.40736753766768044370573974354, −4.49125527143314244312390003472, −3.54067116349947058101732812041, −1.87757341903363932694788708409, 0,
1.87757341903363932694788708409, 3.54067116349947058101732812041, 4.49125527143314244312390003472, 5.40736753766768044370573974354, 6.53745925572367705019854657846, 7.62146689440114469921833710882, 8.430775478361451180134409531171, 9.262817905128557648662036611928, 9.846806652314932081194942554342
