L(s) = 1 | − i·2-s + (−1.5 + 0.866i)3-s − 4-s + (0.866 + 0.5i)5-s + (0.866 + 1.5i)6-s + (2 + 3.46i)7-s + i·8-s + (1.5 − 2.59i)9-s + (0.5 − 0.866i)10-s + (−2.59 + 4.5i)11-s + (1.5 − 0.866i)12-s + (3.46 − 2i)14-s − 1.73·15-s + 16-s + (−2.59 − 4.5i)17-s + (−2.59 − 1.5i)18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.866 + 0.499i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.353 + 0.612i)6-s + (0.755 + 1.30i)7-s + 0.353i·8-s + (0.5 − 0.866i)9-s + (0.158 − 0.273i)10-s + (−0.783 + 1.35i)11-s + (0.433 − 0.249i)12-s + (0.925 − 0.534i)14-s − 0.447·15-s + 0.250·16-s + (−0.630 − 1.09i)17-s + (−0.612 − 0.353i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.413455 + 0.624459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.413455 + 0.624459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-2 + 5.19i)T \) |
good | 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 4.5i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + (4.5 - 2.59i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.19 + 3i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.5 - 4.33i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.3 + 6i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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
−10.18031014761109329840784194159, −9.838576673208879381770884938867, −8.966969436369541695304205814716, −7.937168541592470340787713640206, −6.79214318529036132460915014727, −5.64105892572236102608251823618, −5.07404363898823721767224408476, −4.28828910219472451588349703818, −2.70946714017810608318406937086, −1.81047491374115316195886002699,
0.38965102614282853727009427972, 1.72096346352899877178739416606, 3.71367310408674505092759369359, 4.82059737240067259022960998464, 5.49658953498859178661231458970, 6.41919867791955489009609549501, 7.11623048525930037095570498431, 8.148065679512609473912598858128, 8.481532421166985537213451411702, 10.09668018417691979372102227873