L(s) = 1 | + (1.37 − 2.37i)5-s + (2.64 − 0.0585i)7-s + (−0.771 − 1.33i)11-s + 6.03·13-s + (3.74 + 6.48i)17-s + (−3.01 + 5.22i)19-s + (3.74 − 6.48i)23-s + (−1.27 − 2.20i)25-s − 1.25·29-s + (2.64 + 4.58i)31-s + (3.49 − 6.37i)35-s + (−2.47 + 4.28i)37-s + 5.08·41-s − 3.45·43-s + (−4.74 + 8.22i)47-s + ⋯ |
L(s) = 1 | + (0.614 − 1.06i)5-s + (0.999 − 0.0221i)7-s + (−0.232 − 0.403i)11-s + 1.67·13-s + (0.908 + 1.57i)17-s + (−0.692 + 1.19i)19-s + (0.781 − 1.35i)23-s + (−0.254 − 0.440i)25-s − 0.232·29-s + (0.475 + 0.822i)31-s + (0.590 − 1.07i)35-s + (−0.406 + 0.704i)37-s + 0.794·41-s − 0.527·43-s + (−0.692 + 1.19i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.484995291\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.484995291\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.64 + 0.0585i)T \) |
good | 5 | \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.771 + 1.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 + (-3.74 - 6.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.01 - 5.22i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.74 + 6.48i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 - 4.58i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.47 - 4.28i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + (4.74 - 8.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.91 + 3.32i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.77 + 4.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.29 + 12.6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.01 + 3.49i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 + (6.27 + 10.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.89 - 6.75i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.52T + 83T^{2} \) |
| 89 | \( 1 + (4.74 - 8.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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
−8.738407187222859035369897209805, −8.409356205260453602959559335241, −7.979628634267965291957840741121, −6.42913756934089954864310424192, −5.91843537128015557746777352898, −5.10572298998042804191713488902, −4.27737710529245428659263649229, −3.33338459293055502139585648331, −1.69579144208649132942015013121, −1.22206174304749980329296937364,
1.18524722282997523842063034807, 2.36837212295496969880612972333, 3.18344824037263103341780062431, 4.32121741398335299155693880133, 5.33684006972464517372584738413, 5.95069236909483643033128272788, 7.05846143110844957717309717329, 7.39343404479364501282365025418, 8.500363355580407720428564983627, 9.177924246520236894340195640070