L(s) = 1 | + (−1.52 + 2.58i)3-s + 7.48i·7-s + (−4.32 − 7.89i)9-s − 8.48i·11-s + 10i·13-s + 30.3·17-s + 26.9·19-s + (−19.3 − 11.4i)21-s − 9.17·23-s + (26.9 + 0.905i)27-s + 26.8i·29-s − 8·31-s + (21.9 + 12.9i)33-s + 15.9i·37-s + (−25.8 − 15.2i)39-s + ⋯ |
L(s) = 1 | + (−0.509 + 0.860i)3-s + 1.06i·7-s + (−0.480 − 0.876i)9-s − 0.771i·11-s + 0.769i·13-s + 1.78·17-s + 1.41·19-s + (−0.920 − 0.545i)21-s − 0.398·23-s + (0.999 + 0.0335i)27-s + 0.925i·29-s − 0.258·31-s + (0.663 + 0.393i)33-s + 0.431i·37-s + (−0.661 − 0.392i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.482531906\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.482531906\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.52 - 2.58i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7.48iT - 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 30.3T + 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + 9.17T + 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 15.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 45.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 30.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 26.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.0iT - 9.40e3T^{2} \) |
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
−9.748939431938225921699471690748, −9.185848908517888182517794370168, −8.413135451234823884473083240498, −7.38275845567492122287790134144, −6.14901259767645982076694320352, −5.59822083449066670367393085132, −4.89994570456839981131003422651, −3.61448952250376333798368329562, −2.92011290425297270395796609749, −1.19092333433919579314979078978,
0.55906678386175963430665948398, 1.47856466779378053281949219108, 2.90961931020816535615771411109, 4.02912101898299775868238018810, 5.25358182190563125609191375742, 5.82976337771118473492717497437, 7.02112233109406102646678250942, 7.59923309875272852771835526908, 7.993799972017219232069614218412, 9.467661786170570569853435763037