L(s) = 1 | + 2.09i·5-s + 4.77·7-s + 2.91i·11-s − 6.02i·17-s + 4.35·19-s + 8.99i·23-s + 0.614·25-s + 10.0i·35-s + 10.8·43-s − 13.6i·47-s + 15.8·49-s − 6.10·55-s − 15.1·61-s − 16.8·73-s + 13.9i·77-s + ⋯ |
L(s) = 1 | + 0.936i·5-s + 1.80·7-s + 0.879i·11-s − 1.46i·17-s + 1.00·19-s + 1.87i·23-s + 0.122·25-s + 1.69i·35-s + 1.65·43-s − 1.99i·47-s + 2.26·49-s − 0.823·55-s − 1.94·61-s − 1.96·73-s + 1.58i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.404937756\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.404937756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
good | 5 | \( 1 - 2.09iT - 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 2.91iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6.02iT - 17T^{2} \) |
| 23 | \( 1 - 8.99iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 13.6iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.024431210224107460036236832670, −7.922742132541417218949887833523, −7.35996012018145517498700575027, −7.05382168720677028610688038542, −5.63990486041635276370754726370, −5.10051312411496647433089888011, −4.29902890448271496276615308865, −3.18057909434203012996303894704, −2.23655524628980052958865653295, −1.26898684025014324042785367295,
0.920245764446407294774929266868, 1.67370350017004768666386418413, 2.92039920803672267573712557934, 4.32981354974515683460944309314, 4.60323126173634689588235301001, 5.60208358839565960170291498146, 6.17196468118969760651987564401, 7.52387153089481863999228803428, 8.042823841081682600836472361425, 8.700856533013482378185928879209