L(s) = 1 | + 2.23i·5-s − 13.4·7-s + 17.6i·11-s − 7.48·13-s − 16.9i·17-s − 10.9·19-s + 21.9i·23-s − 5.00·25-s + 47.3i·29-s − 16.9·31-s − 30.1i·35-s − 5.53·37-s − 66.3i·41-s + 38.9·43-s − 32.5i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s − 1.92·7-s + 1.60i·11-s − 0.575·13-s − 0.998i·17-s − 0.577·19-s + 0.952i·23-s − 0.200·25-s + 1.63i·29-s − 0.547·31-s − 0.861i·35-s − 0.149·37-s − 1.61i·41-s + 0.906·43-s − 0.692i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.164999 + 0.519131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164999 + 0.519131i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 + 13.4T + 49T^{2} \) |
| 11 | \( 1 - 17.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.48T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 10.9T + 361T^{2} \) |
| 23 | \( 1 - 21.9iT - 529T^{2} \) |
| 29 | \( 1 - 47.3iT - 841T^{2} \) |
| 31 | \( 1 + 16.9T + 961T^{2} \) |
| 37 | \( 1 + 5.53T + 1.36e3T^{2} \) |
| 41 | \( 1 + 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 38.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76T + 4.48e3T^{2} \) |
| 71 | \( 1 - 77.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 94.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.92T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 124.T + 9.40e3T^{2} \) |
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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
−12.66337127236115120627500544014, −12.15916709214043497211517272927, −10.62776688278013331846767393507, −9.762133443206572667084562312414, −9.177838239533291693781288482266, −7.19075937505196346420280331131, −6.90318336016926118128480497204, −5.39094811154569674040040368375, −3.79106351997745208391470731416, −2.50529463217946527320566340079,
0.30386543257198405174188844239, 2.85437919936228308911185311067, 4.05955638928157510233236492774, 5.89350633994659501334598119124, 6.46754975996789528168434701776, 8.062887184178291795752613879799, 9.067260749798397705614813280084, 9.958034316255844752416700516920, 10.96012719497047626991379481162, 12.28924252438037121365070466984