L(s) = 1 | + (0.342 − 0.939i)3-s + (0.342 − 0.592i)7-s + (−0.766 − 0.642i)9-s + (−0.673 − 1.85i)13-s + (0.866 + 0.5i)19-s + (−0.439 − 0.524i)21-s + (0.939 − 0.342i)25-s + (−0.866 + 0.500i)27-s + (−0.642 + 1.11i)31-s − 1.28i·37-s − 1.96·39-s + (−1.50 + 0.266i)43-s + (0.266 + 0.460i)49-s + (0.766 − 0.642i)57-s + (−1.93 − 0.342i)61-s + ⋯ |
L(s) = 1 | + (0.342 − 0.939i)3-s + (0.342 − 0.592i)7-s + (−0.766 − 0.642i)9-s + (−0.673 − 1.85i)13-s + (0.866 + 0.5i)19-s + (−0.439 − 0.524i)21-s + (0.939 − 0.342i)25-s + (−0.866 + 0.500i)27-s + (−0.642 + 1.11i)31-s − 1.28i·37-s − 1.96·39-s + (−1.50 + 0.266i)43-s + (0.266 + 0.460i)49-s + (0.766 − 0.642i)57-s + (−1.93 − 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.298597678\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.298597678\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.28iT - T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (1.50 - 0.266i)T + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.20 + 1.43i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.642 + 0.233i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
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
−8.271918596707679380256102694092, −7.59760303835244823087084983468, −7.29068008215979505269139159866, −6.30882887045066272128377553260, −5.48272148636990303324253036072, −4.81616751290795791062723670113, −3.44619307536196428122376784041, −2.97586721542883877088998355858, −1.77643571190007507971010581234, −0.71299851389536603876828768568,
1.78044517914422612032922062368, 2.65170638380728273304523296547, 3.55926969501261913142163536020, 4.56598376572146226373181149110, 4.94906160564363291977042908848, 5.84170621138269382005516332178, 6.84112465755662675337344083327, 7.51176986249383452370380406644, 8.564058744628417762563667627097, 8.937595611827864089620894743789