L(s) = 1 | − 0.862i·2-s + 3.07i·3-s + 1.25·4-s + (−1.91 + 1.15i)5-s + 2.65·6-s + 0.566i·7-s − 2.80i·8-s − 6.48·9-s + (1 + 1.64i)10-s − 1.91·11-s + 3.86i·12-s + 0.194i·13-s + 0.488·14-s + (−3.56 − 5.88i)15-s + 0.0877·16-s + 5.29i·17-s + ⋯ |
L(s) = 1 | − 0.610i·2-s + 1.77i·3-s + 0.627·4-s + (−0.855 + 0.518i)5-s + 1.08·6-s + 0.214i·7-s − 0.993i·8-s − 2.16·9-s + (0.316 + 0.521i)10-s − 0.576·11-s + 1.11i·12-s + 0.0539i·13-s + 0.130·14-s + (−0.921 − 1.52i)15-s + 0.0219·16-s + 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4459131514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4459131514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.91 - 1.15i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.862iT - 2T^{2} \) |
| 3 | \( 1 - 3.07iT - 3T^{2} \) |
| 7 | \( 1 - 0.566iT - 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 - 0.194iT - 13T^{2} \) |
| 17 | \( 1 - 5.29iT - 17T^{2} \) |
| 23 | \( 1 + 3.37iT - 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 0.955iT - 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.93iT - 43T^{2} \) |
| 47 | \( 1 - 8.83iT - 47T^{2} \) |
| 53 | \( 1 + 8.20iT - 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 4.04iT - 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 8.60iT - 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 4.51iT - 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00467658659996227203805456844, −9.180223889158059725972491506103, −8.321880676949425259319328470642, −7.51048337249879564308593767014, −6.40896377898622049579682868786, −5.56323232740165298032335213544, −4.51088709982504328602051419925, −3.69401782204706788379362154611, −3.25048684757359274341083092404, −2.15639615218744693726080953915,
0.15207610637883479078034367291, 1.49591775905003944452919833635, 2.47964864564985700282242685080, 3.54610538545572327337898082877, 5.21939632109666124913050948951, 5.63484142696321700653507555999, 6.83025633107633747528277546414, 7.31562544528316238408187253072, 7.63474741725504076093323603774, 8.419326294703815542889287360737