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 | \( 1 + 2.18iT - 97T^{2} \) |
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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
−11.71835223039918664832713241290, −10.64773394093903142654418514403, −10.02524955210246665656122988625, −8.656742853603209863127287921754, −7.45303382238231881949830230136, −6.52079188617424222598819391206, −5.82987009101288303434848899451, −4.65912171807924272651512771495, −3.26916417347108720853729656237, −2.28957179745882660202051676404,
1.67976700410473206441350790468, 2.98926117330532398673005101753, 4.52578541123784101277851868561, 5.22616522637484765938070806832, 6.34543639495012365546543748269, 7.23130498159394045672618810997, 8.622429038391097595415286445580, 9.898430377481666419966523329294, 10.24242431875739008994523394472, 11.62974393478312127335396716085