L(s) = 1 | + 1.82·2-s + 0.105i·3-s + 1.32·4-s − 2.07i·5-s + 0.192i·6-s − 2.92i·7-s − 1.22·8-s + 2.98·9-s − 3.78i·10-s + 0.923i·11-s + 0.140i·12-s − 2.94·13-s − 5.33i·14-s + 0.218·15-s − 4.89·16-s + (−1.45 − 3.85i)17-s + ⋯ |
L(s) = 1 | + 1.29·2-s + 0.0609i·3-s + 0.664·4-s − 0.926i·5-s + 0.0786i·6-s − 1.10i·7-s − 0.433·8-s + 0.996·9-s − 1.19i·10-s + 0.278i·11-s + 0.0404i·12-s − 0.817·13-s − 1.42i·14-s + 0.0564·15-s − 1.22·16-s + (−0.352 − 0.935i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.25498 - 1.56113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25498 - 1.56113i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (1.45 + 3.85i)T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 3 | \( 1 - 0.105iT - 3T^{2} \) |
| 5 | \( 1 + 2.07iT - 5T^{2} \) |
| 7 | \( 1 + 2.92iT - 7T^{2} \) |
| 11 | \( 1 - 0.923iT - 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 0.256iT - 29T^{2} \) |
| 31 | \( 1 - 4.98iT - 31T^{2} \) |
| 37 | \( 1 - 7.01iT - 37T^{2} \) |
| 41 | \( 1 - 6.60iT - 41T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 5.68iT - 61T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 - 9.75iT - 71T^{2} \) |
| 73 | \( 1 - 4.64iT - 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 - 1.66T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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
−10.10296344072879028388533147460, −9.603416383039243045870833878060, −8.480200232863252503051529824728, −7.21882548532120111857394314362, −6.77778329712247778903051449611, −5.19655476322024917332097037226, −4.76404160529440281625347728829, −4.06173649478145580665595121800, −2.82902435507803953266369333151, −1.00558505581928977560913927303,
2.13635174168468992205682039240, 3.15622159708675468843005064370, 4.03274629269907650199773589730, 5.28448870652153578777599738778, 5.81669739741024009931382088250, 6.89988573479828308477350163534, 7.59319514521451920409826626777, 9.009090882167258627916629584773, 9.715968454388855360517096011015, 10.74498948664820196018592916915