L(s) = 1 | + (446. − 576. i)3-s + 1.16e4i·5-s + 2.42e4·7-s + (−1.32e5 − 5.14e5i)9-s + 1.35e6i·11-s − 5.79e6·13-s + (6.70e6 + 5.19e6i)15-s − 4.63e7i·17-s + 3.36e7·19-s + (1.08e7 − 1.39e7i)21-s − 1.86e8i·23-s + 1.08e8·25-s + (−3.55e8 − 1.53e8i)27-s + 2.36e8i·29-s + 5.29e7·31-s + ⋯ |
L(s) = 1 | + (0.612 − 0.790i)3-s + 0.744i·5-s + 0.205·7-s + (−0.248 − 0.968i)9-s + 0.764i·11-s − 1.19·13-s + (0.588 + 0.456i)15-s − 1.92i·17-s + 0.714·19-s + (0.126 − 0.162i)21-s − 1.25i·23-s + 0.445·25-s + (−0.917 − 0.396i)27-s + 0.397i·29-s + 0.0596·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(1.612776751\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612776751\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-446. + 576. i)T \) |
good | 5 | \( 1 - 1.16e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 - 2.42e4T + 1.38e10T^{2} \) |
| 11 | \( 1 - 1.35e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 5.79e6T + 2.32e13T^{2} \) |
| 17 | \( 1 + 4.63e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 3.36e7T + 2.21e15T^{2} \) |
| 23 | \( 1 + 1.86e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 2.36e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 5.29e7T + 7.87e17T^{2} \) |
| 37 | \( 1 + 4.83e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 6.23e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 6.78e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 1.19e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 7.56e9iT - 4.91e20T^{2} \) |
| 59 | \( 1 + 5.06e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 5.11e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 9.11e9T + 8.18e21T^{2} \) |
| 71 | \( 1 - 1.39e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 6.80e9T + 2.29e22T^{2} \) |
| 79 | \( 1 + 3.21e11T + 5.90e22T^{2} \) |
| 83 | \( 1 + 5.38e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 5.38e10iT - 2.46e23T^{2} \) |
| 97 | \( 1 - 7.08e10T + 6.93e23T^{2} \) |
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
−12.58430478030809969028517986751, −11.73140786485483580124006336331, −10.15151941847501005813197630347, −8.983791854760127952955278220814, −7.38285847873404560347430529967, −6.92510922435535049576430673573, −4.97376784916328939470148139548, −3.06282025190465444793469390177, −2.12211715443788773456193289942, −0.39375323377761732490860580526,
1.52704178392593821537966344923, 3.17840489616138815479489636266, 4.49248424043352922792303746202, 5.64378025292469873144710820413, 7.73888531989805003660794583344, 8.726750491806532863798388282522, 9.779582374289739856884442042617, 10.95715163799240930907632092671, 12.37685600300538789554826835372, 13.58327058775724209958350749114