L(s) = 1 | − 2.44i·11-s + 4.35·19-s + 7.34i·23-s − 5·25-s + 10.6i·29-s + 8.71·31-s − 10.6i·41-s − 12.2i·47-s + 7·49-s − 10.6i·53-s + 4·61-s + 8.71·67-s − 8·73-s + 17.4·79-s + 17.1i·83-s + ⋯ |
L(s) = 1 | − 0.738i·11-s + 1.00·19-s + 1.53i·23-s − 25-s + 1.98i·29-s + 1.56·31-s − 1.66i·41-s − 1.78i·47-s + 49-s − 1.46i·53-s + 0.512·61-s + 1.06·67-s − 0.936·73-s + 1.96·79-s + 1.88i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.815136003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.815136003\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 7.34iT - 23T^{2} \) |
| 29 | \( 1 - 10.6iT - 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 17.1iT - 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 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
−8.794380527467231823114581818586, −8.113377213538805520325791473938, −7.28654050907124160503220030161, −6.63478122739025774953993269311, −5.49422116352070136444732504479, −5.23698242596313556070537880208, −3.81307744950846888936935066577, −3.31531652268793683140684995852, −2.07347935549387926473155730649, −0.865023383928781860127734522779,
0.838750504355816505044282296078, 2.19858349996968391383117118224, 3.00292553853139417312288190301, 4.30660657577960647378903709756, 4.67062556177128232334966462633, 5.94166324995625963726261382647, 6.38705163435327954827334765663, 7.49489567655296665223259594414, 7.931595683593498849457518172275, 8.810956086850532206342820881662