L(s) = 1 | + (−0.195 − 0.980i)2-s + (−0.923 + 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.555 + 0.831i)8-s + i·10-s + (0.707 − 0.707i)16-s + (−0.275 − 0.275i)17-s + (0.216 + 1.08i)19-s + (0.980 − 0.195i)20-s + (0.360 − 0.149i)23-s + (0.923 + 0.382i)25-s + 0.765·31-s + (−0.831 − 0.555i)32-s + (−0.216 + 0.324i)34-s + (1.02 − 0.425i)38-s + ⋯ |
L(s) = 1 | + (−0.195 − 0.980i)2-s + (−0.923 + 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.555 + 0.831i)8-s + i·10-s + (0.707 − 0.707i)16-s + (−0.275 − 0.275i)17-s + (0.216 + 1.08i)19-s + (0.980 − 0.195i)20-s + (0.360 − 0.149i)23-s + (0.923 + 0.382i)25-s + 0.765·31-s + (−0.831 − 0.555i)32-s + (−0.216 + 0.324i)34-s + (1.02 − 0.425i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8018862565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8018862565\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.195 + 0.980i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.980 + 0.195i)T \) |
good | 7 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (0.275 + 0.275i)T + iT^{2} \) |
| 19 | \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.360 + 0.149i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 - 0.765T + T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + (-1.17 + 1.17i)T - iT^{2} \) |
| 53 | \( 1 + (0.425 + 0.636i)T + (-0.382 + 0.923i)T^{2} \) |
| 59 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 61 | \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-1 + i)T - iT^{2} \) |
| 83 | \( 1 + (-1.81 + 0.360i)T + (0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−8.869336062574182283019210970589, −8.135244695153436842031062955541, −7.62760971245097987987950001708, −6.65449654519119341680661283459, −5.44359022414734031515040102487, −4.67345077170989506065767916545, −3.88159127652205905831468340455, −3.21283502110803148818852562927, −2.11433052639491646427543797120, −0.811172024376363920831962258595,
0.894488164158395600077355902703, 2.69291977062075352942796685570, 3.81154109653792027847708222960, 4.50706186717828166568298883043, 5.27103380211414706504728466675, 6.25837216930343738150295412996, 6.98042699816197102102497087589, 7.51566792098905510438031627027, 8.282450080752603313108335935888, 8.889096743814216807305013645847