L(s) = 1 | + (−0.232 − 1.71i)3-s + (−0.581 − 0.335i)5-s + (−2.63 + 0.209i)7-s + (−2.89 + 0.798i)9-s + (−2.62 − 4.54i)11-s + 2·13-s + (−0.440 + 1.07i)15-s + (−3.64 + 2.10i)17-s + (−1.13 − 0.656i)19-s + (0.972 + 4.47i)21-s + (1.60 − 2.77i)23-s + (−2.27 − 3.94i)25-s + (2.04 + 4.77i)27-s + 6.22i·29-s + (−1.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + (−0.134 − 0.990i)3-s + (−0.259 − 0.150i)5-s + (−0.996 + 0.0791i)7-s + (−0.963 + 0.266i)9-s + (−0.791 − 1.37i)11-s + 0.554·13-s + (−0.113 + 0.277i)15-s + (−0.884 + 0.510i)17-s + (−0.260 − 0.150i)19-s + (0.212 + 0.977i)21-s + (0.334 − 0.578i)23-s + (−0.454 − 0.788i)25-s + (0.393 + 0.919i)27-s + 1.15i·29-s + (−0.269 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0830497 - 0.595168i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0830497 - 0.595168i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.232 + 1.71i)T \) |
| 7 | \( 1 + (2.63 - 0.209i)T \) |
good | 5 | \( 1 + (0.581 + 0.335i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.62 + 4.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3.64 - 2.10i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.13 + 0.656i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.60 + 2.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.22iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.13 + 5.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.76iT - 41T^{2} \) |
| 43 | \( 1 + 9.55iT - 43T^{2} \) |
| 47 | \( 1 + (4.80 - 8.32i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.1 + 5.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.78 - 6.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 3.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.58 + 1.49i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.08T + 71T^{2} \) |
| 73 | \( 1 + (0.137 + 0.238i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 + 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.04T + 83T^{2} \) |
| 89 | \( 1 + (12.1 + 6.98i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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
−11.09618860105911282113969805420, −10.52141806187396004609625793980, −8.848243734296758293167220079702, −8.455404609031794591755728206468, −7.18662555806489662656741387529, −6.29406735634329154066807941326, −5.50018792396658723319498696938, −3.69460534970556405918286296509, −2.46796288220204244113569621819, −0.40497073075807073476602007324,
2.67299976355912940572491919873, 3.89282715606735672880791422504, 4.87516767675835353481689003641, 6.08715602589447382149935984639, 7.15021791808382270821987902218, 8.329617905773931276652496754453, 9.646702084490175626337367677158, 9.837540240076210170443867249958, 11.02821487904571660410816516189, 11.73363700806735089145682137197