L(s) = 1 | + (−2.09 − 3.62i)5-s + (2.64 + 0.0532i)7-s + (−1.23 − 0.711i)11-s + (0.850 − 0.491i)13-s + 0.370·17-s − 4.97i·19-s + (4.98 − 2.87i)23-s + (−6.26 + 10.8i)25-s + (−7.31 − 4.22i)29-s + (6.28 − 3.62i)31-s + (−5.34 − 9.70i)35-s + 3.46·37-s + (−1.06 − 1.85i)41-s + (−3.00 + 5.21i)43-s + (4.13 − 7.16i)47-s + ⋯ |
L(s) = 1 | + (−0.936 − 1.62i)5-s + (0.999 + 0.0201i)7-s + (−0.371 − 0.214i)11-s + (0.235 − 0.136i)13-s + 0.0899·17-s − 1.14i·19-s + (1.04 − 0.600i)23-s + (−1.25 + 2.17i)25-s + (−1.35 − 0.784i)29-s + (1.12 − 0.651i)31-s + (−0.903 − 1.64i)35-s + 0.569·37-s + (−0.167 − 0.289i)41-s + (−0.458 + 0.794i)43-s + (0.603 − 1.04i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.257751890\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257751890\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.64 - 0.0532i)T \) |
good | 5 | \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.23 + 0.711i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.850 + 0.491i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.370T + 17T^{2} \) |
| 19 | \( 1 + 4.97iT - 19T^{2} \) |
| 23 | \( 1 + (-4.98 + 2.87i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.31 + 4.22i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.28 + 3.62i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 + (1.06 + 1.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.00 - 5.21i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.97iT - 53T^{2} \) |
| 59 | \( 1 + (-2.27 - 3.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.50 + 3.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.03 + 8.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + 9.52iT - 73T^{2} \) |
| 79 | \( 1 + (4.25 - 7.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.972 - 1.68i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 + (-3.34 - 1.92i)T + (48.5 + 84.0i)T^{2} \) |
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
−8.383206373350103326005944202631, −7.85085031375370219071637987139, −7.17784055049646987281608436243, −5.89080963941075187548275168252, −5.08614743658981699071858011978, −4.59817817895417855645689376810, −3.91079159010438384702860351731, −2.61452454765311211121627089108, −1.31187155333133311222775405115, −0.42916984662637562173461469756,
1.50302582506875788749245059018, 2.67112750112691655070852446345, 3.46211491778554364075460823891, 4.20824811027408156131867683068, 5.19378732684056262913143780481, 6.10903258093689001304452242439, 7.01132678891147100038374410492, 7.53123369046303974683533691248, 8.053391865786299173233066910648, 8.887815786855257840838589819021