L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.295 − 2.62i)7-s + 0.999i·8-s + (−0.570 − 0.329i)11-s + 6.13i·13-s + (1.05 + 2.42i)14-s + (−0.5 − 0.866i)16-s + (2.43 − 4.22i)17-s + (−6.30 + 3.63i)19-s + 0.659·22-s + (3.98 − 2.29i)23-s + (−3.06 − 5.31i)26-s + (−2.12 − 1.57i)28-s − 8.09i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.111 − 0.993i)7-s + 0.353i·8-s + (−0.172 − 0.0993i)11-s + 1.70i·13-s + (0.282 + 0.648i)14-s + (−0.125 − 0.216i)16-s + (0.591 − 1.02i)17-s + (−1.44 + 0.834i)19-s + 0.140·22-s + (0.830 − 0.479i)23-s + (−0.601 − 1.04i)26-s + (−0.402 − 0.296i)28-s − 1.50i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.273084046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.273084046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.295 + 2.62i)T \) |
good | 11 | \( 1 + (0.570 + 0.329i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.13iT - 13T^{2} \) |
| 17 | \( 1 + (-2.43 + 4.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.30 - 3.63i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.98 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.09iT - 29T^{2} \) |
| 31 | \( 1 + (-0.759 - 0.438i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.05 - 8.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.25T + 41T^{2} \) |
| 43 | \( 1 - 9.03T + 43T^{2} \) |
| 47 | \( 1 + (-6.00 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.5 + 6.10i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.06 - 7.04i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0618 - 0.0357i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.666 + 1.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.60iT - 71T^{2} \) |
| 73 | \( 1 + (-2.44 - 1.41i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.88 + 5.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + (-2.66 - 4.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.4iT - 97T^{2} \) |
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
−8.632791840830149857300768709261, −7.84340861854303794537389704409, −7.30247738271510129096749293921, −6.47819056314328764900662122418, −5.97833829425160822481417163807, −4.48717664046099572489759286710, −4.37592879688725162326963210909, −2.91238642240563876297584887504, −1.82111839479754547435477399955, −0.69907694863716384530186491817,
0.805919120195389238158816506815, 2.12436793534501079931328348604, 2.85797718412954240346044931230, 3.74522402212648494804858469544, 4.95290471747822652557286192976, 5.69419739364720033001936040936, 6.38681762706310997125445948601, 7.53118243072806184182047587648, 7.973523407698493804644868057015, 8.919293818958617462676766116230