L(s) = 1 | + (0.5 − 0.866i)3-s + (2 − 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 2·13-s + (1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s + (−0.499 − 2.59i)21-s + (1.5 + 2.59i)23-s + 5·27-s − 6·29-s + (3.5 − 6.06i)31-s + (−1.5 − 2.59i)33-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 0.554·13-s + (0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s + (−0.109 − 0.566i)21-s + (0.312 + 0.541i)23-s + 0.962·27-s − 1.11·29-s + (0.628 − 1.08i)31-s + (−0.261 − 0.452i)33-s + (−0.0821 − 0.142i)37-s + (−0.160 + 0.277i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66147 - 0.823639i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66147 - 0.823639i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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
−10.35508363992139003417916379326, −9.458114382926935858904750426256, −8.427759257669920933694000067125, −7.59618411399666916661370408777, −7.15087059010579632517495786401, −5.81820520540124750831627349732, −4.82575660321742850830212789460, −3.76672952584091699620379785258, −2.38062798038959765339881574384, −1.10346059572545146215595998039,
1.59219404940058172339480430264, 2.94732356143077770675647655073, 4.21252357604103652490734770425, 4.92643690093785376021988999200, 6.10186939289148981230781709907, 7.12794565626510936920685402125, 8.068373590334878660937031328481, 9.045154973994390634652297060305, 9.563724693824625332199937660980, 10.48901972068012868972455417702