L(s) = 1 | + (1.37 − 0.331i)2-s + (1.78 − 0.910i)4-s + (1.64 − 1.51i)5-s − 0.936·7-s + (2.14 − 1.84i)8-s + (1.76 − 2.62i)10-s + 2.20i·11-s − 3.33·13-s + (−1.28 + 0.310i)14-s + (2.34 − 3.24i)16-s + 1.54·17-s − 3.12·19-s + (1.56 − 4.19i)20-s + (0.731 + 3.03i)22-s + 3.39i·23-s + ⋯ |
L(s) = 1 | + (0.972 − 0.234i)2-s + (0.890 − 0.455i)4-s + (0.737 − 0.675i)5-s − 0.353·7-s + (0.759 − 0.650i)8-s + (0.558 − 0.829i)10-s + 0.665i·11-s − 0.924·13-s + (−0.344 + 0.0828i)14-s + (0.585 − 0.810i)16-s + 0.374·17-s − 0.716·19-s + (0.349 − 0.937i)20-s + (0.155 + 0.647i)22-s + 0.707i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.34975 - 0.905818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.34975 - 0.905818i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 + 0.331i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.64 + 1.51i)T \) |
good | 7 | \( 1 + 0.936T + 7T^{2} \) |
| 11 | \( 1 - 2.20iT - 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 3.39iT - 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 8.30iT - 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 + 7.77iT - 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 5.59iT - 73T^{2} \) |
| 79 | \( 1 - 1.02iT - 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.18iT - 97T^{2} \) |
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
−11.62974393478312127335396716085, −10.24242431875739008994523394472, −9.898430377481666419966523329294, −8.622429038391097595415286445580, −7.23130498159394045672618810997, −6.34543639495012365546543748269, −5.22616522637484765938070806832, −4.52578541123784101277851868561, −2.98926117330532398673005101753, −1.67976700410473206441350790468,
2.28957179745882660202051676404, 3.26916417347108720853729656237, 4.65912171807924272651512771495, 5.82987009101288303434848899451, 6.52079188617424222598819391206, 7.45303382238231881949830230136, 8.656742853603209863127287921754, 10.02524955210246665656122988625, 10.64773394093903142654418514403, 11.71835223039918664832713241290