L(s) = 1 | + 10.7i·3-s + 198.·7-s + 127.·9-s − 85.9i·11-s − 407. i·13-s − 1.20e3·17-s + 206. i·19-s + 2.13e3i·21-s − 2.59e3·23-s + 3.98e3i·27-s + 6.19e3i·29-s + 1.86e3·31-s + 923.·33-s + 1.47e4i·37-s + 4.37e3·39-s + ⋯ |
L(s) = 1 | + 0.689i·3-s + 1.53·7-s + 0.524·9-s − 0.214i·11-s − 0.668i·13-s − 1.01·17-s + 0.130i·19-s + 1.05i·21-s − 1.02·23-s + 1.05i·27-s + 1.36i·29-s + 0.348·31-s + 0.147·33-s + 1.76i·37-s + 0.460·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.589610622\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.589610622\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 10.7iT - 243T^{2} \) |
| 7 | \( 1 - 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.9iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 407. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 206. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.19e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 6.26e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.80e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.829055760924839840032225536085, −8.805998878001819002726515998249, −8.126612767766780811784137957575, −7.30192435106135733962752127582, −6.12846804399562366202695169181, −4.96777627182504535229515008111, −4.55062728153249575166220465507, −3.45505196578484690002281938467, −2.07626578319971181913713868359, −1.07930580398720528151792794064,
0.55134763466201568421025734177, 1.77897242583659381088160207347, 2.22312843097492048754441166216, 4.15293269760839087867100600976, 4.61901109420325951015323905405, 5.89794878421527816848210334329, 6.80634066049668857049592518242, 7.68988533834901448899621329340, 8.194924577800514045424429423428, 9.237843633260651198688089136736