L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (0.5 − 2.59i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s − 5·13-s + (2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (−0.499 − 0.866i)18-s + (3.5 − 6.06i)19-s + (2.5 − 0.866i)21-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (0.188 − 0.981i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.144 − 0.249i)12-s − 1.38·13-s + (0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.117 − 0.204i)18-s + (0.802 − 1.39i)19-s + (0.545 − 0.188i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7972185887\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7972185887\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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
−9.525766818303533896474423004274, −9.204933924759514965685025924644, −7.87149881954552970509503110321, −7.38364876412076295594692138021, −6.67215301209161030783520210664, −5.18095264175520696050491607986, −4.77488490657734986152979262426, −3.59435046012128353016069701723, −2.26253071491922606438835554724, −0.37162367235802753064765556360,
1.67875849839359839785987381729, 2.45169380120769233031150109519, 3.57009651550726298748689200091, 4.79711603892737432456889712207, 5.82521485263371149903106564716, 6.81607552527801770307819077713, 7.88753918827989874547607118196, 8.405253032360885644011994647749, 9.239803040726290765750313419768, 10.02703468974041877185251715036