L(s) = 1 | − 2.23·5-s + 4.33i·7-s − 3·9-s + 7.50·17-s + 4.79i·23-s + 5.00·25-s − 8.78·29-s − 2.76i·31-s − 9.69i·35-s − 1.43·37-s − 12.7·41-s − 9.59i·43-s + 6.70·45-s − 11.7·49-s − 13.5·53-s + ⋯ |
L(s) = 1 | − 0.999·5-s + 1.63i·7-s − 9-s + 1.82·17-s + 0.999i·23-s + 1.00·25-s − 1.63·29-s − 0.496i·31-s − 1.63i·35-s − 0.236·37-s − 1.99·41-s − 1.46i·43-s + 0.999·45-s − 1.68·49-s − 1.86·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1285857743\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1285857743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 23 | \( 1 - 4.79iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 4.33iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 9.59iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 4.16iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 16.6iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−9.398736734654701117848785989394, −8.968479342789337542708535322553, −8.028834519904832576569000049702, −7.68741026226108527760165205045, −6.41858993976520197176143472666, −5.48097170562984408667543455825, −5.18311212328645706816281503856, −3.56936192007090404864714354983, −3.12659739086198694620414617399, −1.85031029663865860249947389488,
0.05081423824520979663742650248, 1.24964154119273593094300633008, 3.11670386896734681067569997534, 3.62539720177268157900757344410, 4.56806734117829192022300143204, 5.46137886640543427159445608280, 6.58762037294556506811263766663, 7.34503458368562443340299281309, 7.961611729286927618790035466928, 8.535578617439334991464627574412