L(s) = 1 | + (−1.20 − 0.439i)3-s + (0.500 + 0.419i)9-s + (−1.62 + 0.939i)11-s + (−0.766 + 0.642i)17-s + (0.984 + 0.173i)19-s + (−0.939 + 0.342i)25-s + (0.223 + 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (−0.984 + 0.173i)43-s + (0.5 + 0.866i)49-s + (1.20 − 0.439i)51-s + (−1.11 − 0.642i)57-s + (−0.524 + 0.439i)59-s + (1.50 + 1.26i)67-s + ⋯ |
L(s) = 1 | + (−1.20 − 0.439i)3-s + (0.500 + 0.419i)9-s + (−1.62 + 0.939i)11-s + (−0.766 + 0.642i)17-s + (0.984 + 0.173i)19-s + (−0.939 + 0.342i)25-s + (0.223 + 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (−0.984 + 0.173i)43-s + (0.5 + 0.866i)49-s + (1.20 − 0.439i)51-s + (−1.11 − 0.642i)57-s + (−0.524 + 0.439i)59-s + (1.50 + 1.26i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3212238297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3212238297\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.984 - 0.173i)T \) |
good | 3 | \( 1 + (1.20 + 0.439i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (0.524 - 0.439i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
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
−10.24661381025086247158080376323, −9.622765967985312224275333531184, −8.337221969868430971957134398823, −7.56937593852444007676953475288, −6.84013551877478630278207164182, −5.89679754881924424947102108435, −5.24761303976341749499547034582, −4.42894924971301466080474414033, −2.92694624277638073619558766126, −1.62152191060400622263789908734,
0.31488439567298742981317308735, 2.41509851765868934369956338044, 3.60077786151607904759877640580, 4.92489450380193893050834668700, 5.34111807243965903564368044957, 6.11980299571529869837845191298, 7.14483368964611273997087926743, 8.041187890251214886720557221934, 8.913319659444259128385064512110, 10.03774073401076988068217335169