L(s) = 1 | + (−1 + 1.73i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + 2·13-s + (3 + 5.19i)17-s + (4 − 6.92i)19-s + (0.500 + 0.866i)25-s + 6·29-s + (4 + 6.92i)31-s + (1 − 1.73i)37-s + 2·41-s + 4·43-s + (3 + 5.19i)45-s + (−4 + 6.92i)47-s + (−3 − 5.19i)53-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + 0.554·13-s + (0.727 + 1.26i)17-s + (0.917 − 1.58i)19-s + (0.100 + 0.173i)25-s + 1.11·29-s + (0.718 + 1.24i)31-s + (0.164 − 0.284i)37-s + 0.312·41-s + 0.609·43-s + (0.447 + 0.774i)45-s + (−0.583 + 1.01i)47-s + (−0.412 − 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50020 - 0.0952033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50020 - 0.0952033i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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
−10.48294625459876089004200459032, −9.425685092397463660603595718644, −8.530350405450584546386585811456, −7.69809448511971068841847065482, −6.73519707138162428312821304226, −6.07202499312348906035746184998, −4.84894503885296585324137864829, −3.52774099205285981963357840676, −2.99821593164513404316730834034, −0.995768832457415093724212806481,
1.17615431779859805531239590130, 2.62212505602142897624953553630, 4.08748519784175020144851615366, 4.87244005875456822148819751026, 5.67216051759803592901494515599, 7.08398838949866741390065546443, 7.85031518978284808413217539851, 8.354469909226327580269052273759, 9.781253468500353973831116709537, 10.00105542913798980216577191462