L(s) = 1 | + (0.309 + 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.978 − 0.207i)7-s + (−0.809 + 0.587i)9-s + (−0.978 + 0.207i)12-s + (0.139 − 0.155i)13-s + (−0.978 − 0.207i)16-s + (−0.244 − 1.14i)19-s + (0.5 + 0.866i)21-s − 25-s + (−0.809 − 0.587i)27-s + (0.104 + 0.994i)28-s + (0.978 + 0.207i)31-s + (−0.5 − 0.866i)36-s + (0.360 − 0.207i)37-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.978 − 0.207i)7-s + (−0.809 + 0.587i)9-s + (−0.978 + 0.207i)12-s + (0.139 − 0.155i)13-s + (−0.978 − 0.207i)16-s + (−0.244 − 1.14i)19-s + (0.5 + 0.866i)21-s − 25-s + (−0.809 − 0.587i)27-s + (0.104 + 0.994i)28-s + (0.978 + 0.207i)31-s + (−0.5 − 0.866i)36-s + (0.360 − 0.207i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 651 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0712 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 651 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0712 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.044683571\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.044683571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.978 + 0.207i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
good | 2 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.139 + 0.155i)T + (-0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.244 + 1.14i)T + (-0.913 + 0.406i)T^{2} \) |
| 23 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 29 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.360 + 0.207i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (1.41 - 1.27i)T + (0.104 - 0.994i)T^{2} \) |
| 47 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 53 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.913 + 1.58i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 73 | \( 1 + (-1.22 + 0.544i)T + (0.669 - 0.743i)T^{2} \) |
| 79 | \( 1 + (0.809 - 1.81i)T + (-0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 97 | \( 1 + (-0.478 + 0.658i)T + (-0.309 - 0.951i)T^{2} \) |
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
−11.22180334252576557816064761803, −10.02409867731497559930817156826, −9.177548355459874317603935904609, −8.261851359591480735863448498184, −7.87687308583818381423146417432, −6.61531709302805411923530354983, −5.12575272077948439768520545609, −4.44855560745142449581421036003, −3.49673447100127983341624560276, −2.36188041589498580223174952639,
1.37858950131948021447132864807, 2.30491786140114385542705935895, 4.02534695338517359404518679576, 5.31696581219953946546798799687, 6.03224103998701505170754323822, 6.97881255883504486203968298842, 8.057090326519763462067969050998, 8.622730396672875506249057177463, 9.699554673002377297498708648991, 10.54028241987996932042008472162