L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−1.62 − 0.781i)5-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (−1.40 + 1.12i)10-s + (−0.0990 + 0.433i)13-s + (−0.222 + 0.974i)16-s + 1.24i·17-s + (0.974 + 0.222i)18-s + (0.400 + 1.75i)20-s + (1.40 + 1.75i)25-s + (0.347 + 0.277i)26-s + (0.781 + 0.623i)32-s + (1.12 + 0.541i)34-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−1.62 − 0.781i)5-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (−1.40 + 1.12i)10-s + (−0.0990 + 0.433i)13-s + (−0.222 + 0.974i)16-s + 1.24i·17-s + (0.974 + 0.222i)18-s + (0.400 + 1.75i)20-s + (1.40 + 1.75i)25-s + (0.347 + 0.277i)26-s + (0.781 + 0.623i)32-s + (1.12 + 0.541i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7517882706\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7517882706\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.433 + 0.900i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 5 | \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 - 1.24iT - T^{2} \) |
| 19 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 37 | \( 1 + (-0.433 + 0.0990i)T + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 - 0.445iT - T^{2} \) |
| 43 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.974 - 0.777i)T + (0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.541 - 1.12i)T + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.781 - 1.62i)T + (-0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 + (0.347 - 0.277i)T + (0.222 - 0.974i)T^{2} \) |
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
−8.636694860332529708169417703968, −8.263575498064712712422865456816, −7.52259018483026214626837279678, −6.50594639542404330429569322039, −5.37663744763047369690432546093, −4.73610095691319157779329510624, −4.08606010342060643044127036567, −3.51288000956145562580553789301, −2.26244850559362528556784009876, −1.18726022431312358912111132580,
0.44659596654065465258363951561, 2.90092970011938901808257011567, 3.36950751963046968860643793382, 4.20906832202077427941654265193, 4.84076676614358162106228339010, 5.97178768618168464220830508529, 6.71775228212666523399159297252, 7.31426485680792718476624892598, 7.76205344180992858778744371074, 8.549814982279750264679036191881