L(s) = 1 | + 3.46·7-s − 6.92i·13-s + 8i·19-s + 5·25-s + 10.3·31-s − 6.92i·37-s − 8i·43-s + 4.99·49-s + 6.92i·61-s + 16i·67-s − 10·73-s − 17.3·79-s − 23.9i·91-s − 14·97-s − 3.46·103-s + ⋯ |
L(s) = 1 | + 1.30·7-s − 1.92i·13-s + 1.83i·19-s + 25-s + 1.86·31-s − 1.13i·37-s − 1.21i·43-s + 0.714·49-s + 0.887i·61-s + 1.95i·67-s − 1.17·73-s − 1.94·79-s − 2.51i·91-s − 1.42·97-s − 0.341·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67336 - 0.220302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67336 - 0.220302i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.47415919513881945463296667347, −10.18582054018850865422955820575, −8.626616647088873930807609418283, −8.113253922973338163478279582220, −7.34041290884689781296131653410, −5.88784431602400786038619900080, −5.22282850136318536068324803627, −4.06811425713124686510116892676, −2.74867884631687247760772303944, −1.21724142130071695066294077554,
1.44149313951259936804694043266, 2.74081942433013216509787079475, 4.57432985796372147118514508149, 4.74496493096560457889999460391, 6.39888169397866435824480148210, 7.09600411450398181081669081802, 8.241632888835450592593425866426, 8.918115939291970535912697845731, 9.805053005801765873574366668706, 11.10521143790207491460421182802