L(s) = 1 | − 0.601i·5-s + 0.163i·11-s − 3.37i·13-s − 2.13i·17-s + 0.953·19-s + 6.99i·23-s + 4.63·25-s + 1.28·29-s + 3.35·31-s + 5.16·37-s + 6.67i·41-s + 6.09i·43-s + 3.69·47-s − 13.8·53-s + 0.0984·55-s + ⋯ |
L(s) = 1 | − 0.269i·5-s + 0.0493i·11-s − 0.937i·13-s − 0.517i·17-s + 0.218·19-s + 1.45i·23-s + 0.927·25-s + 0.239·29-s + 0.602·31-s + 0.848·37-s + 1.04i·41-s + 0.928i·43-s + 0.539·47-s − 1.90·53-s + 0.0132·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.996939082\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.996939082\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.601iT - 5T^{2} \) |
| 11 | \( 1 - 0.163iT - 11T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 - 0.953T + 19T^{2} \) |
| 23 | \( 1 - 6.99iT - 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 - 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 6.09iT - 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 + 0.181iT - 61T^{2} \) |
| 67 | \( 1 + 9.09iT - 67T^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.98iT - 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 + 9.08iT - 97T^{2} \) |
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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
−7.80649088133899960272652225914, −7.40408946202599784042828305660, −6.39810181134706116372147958417, −5.83754389710701760311248859526, −4.95826817954769130258252865469, −4.52374748929894886388019090251, −3.29002926385393879999447321126, −2.89976499777154107160782332529, −1.61835957062790827618204699634, −0.70359030827357003728864488301,
0.76585651292532547125401732258, 1.94879015211767594670219757309, 2.72223944374069305320843997998, 3.63298854217707644314815530185, 4.45267272867756936744329381128, 5.01008031949873976884973656927, 6.12956485605784970978475878775, 6.49274836446369399932703167273, 7.23504686601505578187467051657, 7.980269660439416769533819790716