L(s) = 1 | + (−1.88 − 2.11i)2-s + (−0.909 + 7.94i)4-s + 21.8i·5-s + 7i·7-s + (18.4 − 13.0i)8-s + (46.1 − 41.1i)10-s + 66.1·11-s − 61.5·13-s + (14.7 − 13.1i)14-s + (−62.3 − 14.4i)16-s + 11.7i·17-s + 87.9i·19-s + (−173. − 19.8i)20-s + (−124. − 139. i)22-s − 13.2·23-s + ⋯ |
L(s) = 1 | + (−0.665 − 0.746i)2-s + (−0.113 + 0.993i)4-s + 1.95i·5-s + 0.377i·7-s + (0.817 − 0.576i)8-s + (1.45 − 1.30i)10-s + 1.81·11-s − 1.31·13-s + (0.282 − 0.251i)14-s + (−0.974 − 0.225i)16-s + 0.167i·17-s + 1.06i·19-s + (−1.94 − 0.222i)20-s + (−1.20 − 1.35i)22-s − 0.119·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9178953605\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9178953605\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.88 + 2.11i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 21.8iT - 125T^{2} \) |
| 11 | \( 1 - 66.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 87.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 13.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.53iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 45.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 172. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 331. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 282. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 553.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 262.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.8iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 891.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 836. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 114.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 162. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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
−11.73077110328869157575413930065, −10.97328092436583200272061373255, −9.964753440137997818282029003631, −9.441690937604493768553465832687, −7.976797089671287718752116957905, −7.05392507257029502683407945407, −6.23595217634024140895788167523, −4.05804735869043749290119008158, −3.05071158834713164255950909823, −1.90902172095726444394062809823,
0.46847667431009834038721967443, 1.55887517846684526656917793325, 4.37433884902142351592499331579, 5.00485679950224298333238559450, 6.32727136599479896308058284956, 7.44499074491870631515135953587, 8.480395490051854057518651007446, 9.340342246585167869872983743856, 9.681148190904964222185125600534, 11.35926861507312724438802341366