| 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
−8.535578617439334991464627574412, −7.961611729286927618790035466928, −7.34503458368562443340299281309, −6.58762037294556506811263766663, −5.46137886640543427159445608280, −4.56806734117829192022300143204, −3.62539720177268157900757344410, −3.11670386896734681067569997534, −1.24964154119273593094300633008, −0.05081423824520979663742650248,
1.85031029663865860249947389488, 3.12659739086198694620414617399, 3.56936192007090404864714354983, 5.18311212328645706816281503856, 5.48097170562984408667543455825, 6.41858993976520197176143472666, 7.68741026226108527760165205045, 8.028834519904832576569000049702, 8.968479342789337542708535322553, 9.398736734654701117848785989394
