L(s) = 1 | + (−0.281 − 0.959i)2-s + (0.909 − 0.415i)3-s + (−0.841 + 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.654 − 0.755i)6-s + (0.755 + 0.654i)8-s + (0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.540 + 0.841i)12-s + (−0.415 + 0.909i)15-s + (0.415 − 0.909i)16-s + (−0.540 − 1.84i)17-s + (−0.909 − 0.415i)18-s + (1.61 + 0.474i)19-s + (0.281 − 0.959i)20-s + ⋯ |
L(s) = 1 | + (−0.281 − 0.959i)2-s + (0.909 − 0.415i)3-s + (−0.841 + 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.654 − 0.755i)6-s + (0.755 + 0.654i)8-s + (0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.540 + 0.841i)12-s + (−0.415 + 0.909i)15-s + (0.415 − 0.909i)16-s + (−0.540 − 1.84i)17-s + (−0.909 − 0.415i)18-s + (1.61 + 0.474i)19-s + (0.281 − 0.959i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.084692185\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.084692185\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.281 + 0.959i)T \) |
| 3 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-0.989 + 0.142i)T \) |
good | 7 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 11 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 17 | \( 1 + (0.540 + 1.84i)T + (-0.841 + 0.540i)T^{2} \) |
| 19 | \( 1 + (-1.61 - 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (-0.983 - 0.449i)T + (0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 41 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 - 0.284iT - T^{2} \) |
| 53 | \( 1 + (0.153 - 0.239i)T + (-0.415 - 0.909i)T^{2} \) |
| 59 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2} \) |
| 67 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 79 | \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \) |
| 89 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
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
−9.480797793395062916131865303600, −8.993028578006646229681075420184, −7.952079182927239970237016859086, −7.46606980574528824228113518125, −6.72083660248685957136073671020, −5.05980152384119056135197618614, −4.12676938640941999816300404661, −3.00444966527152211315497775845, −2.77268044837545494908524433426, −1.12065922916840325248354440238,
1.41623183117356523057394342628, 3.22695799640395702221237327121, 4.17423366886745734735417092819, 4.82668925685584031497380628338, 5.79562101731875282496058174820, 7.03412884272454944192865887117, 7.65498045557211212786656721901, 8.443877373422758583666371874205, 8.856092884869174486417191737798, 9.650217728287567148285695687082