| L(s) = 1 | + 2i·2-s − 1.62i·3-s − 4·4-s + 3.24·6-s + 2.62i·7-s − 8i·8-s + 24.3·9-s − 51.1·11-s + 6.49i·12-s − 13i·13-s − 5.24·14-s + 16·16-s + 4.36i·17-s + 48.7i·18-s + 47.4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.312i·3-s − 0.5·4-s + 0.220·6-s + 0.141i·7-s − 0.353i·8-s + 0.902·9-s − 1.40·11-s + 0.156i·12-s − 0.277i·13-s − 0.100·14-s + 0.250·16-s + 0.0622i·17-s + 0.638i·18-s + 0.573·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.788323963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788323963\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 13iT \) |
| good | 3 | \( 1 + 1.62iT - 27T^{2} \) |
| 7 | \( 1 - 2.62iT - 343T^{2} \) |
| 11 | \( 1 + 51.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 4.36iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 47.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 377. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 7.71T + 6.89e4T^{2} \) |
| 43 | \( 1 + 75.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 186. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 236. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 4.36T + 2.05e5T^{2} \) |
| 61 | \( 1 - 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 98.2iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 404. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 856.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 920. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 960. iT - 9.12e5T^{2} \) |
| 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
−10.11260102476124476804925987694, −9.319338338000401015657113912916, −8.158542131109715790041623330803, −7.59159891177575733982919487694, −6.80618862138402083258021255408, −5.66334557253068368440901373465, −4.95734142805661577683190918634, −3.72395545287353145770519826938, −2.33149936289840854596807597285, −0.75220340419365490380966276786,
0.830669908655800126291687768571, 2.25631649111312587671975089838, 3.33159046459911736920967532245, 4.52452948077747458748486351323, 5.14699670409903873733809223291, 6.52883180508908881034571198945, 7.60572993251976460974912084798, 8.431060422507145641270022636387, 9.504625471068891837142373137078, 10.26949456076721477479630390336
