| L(s) = 1 | + 4.16·5-s − 2.49·7-s − 11-s − 6.89·13-s − 1.55·17-s − 6.43·19-s + 6.27·23-s + 12.3·25-s + 4.83·29-s + 0.425·31-s − 10.3·35-s − 10.5·37-s − 1.30·41-s − 4.36·43-s − 8.92·47-s − 0.773·49-s + 3.84·53-s − 4.16·55-s − 6.93·59-s + 5.26·61-s − 28.7·65-s − 5.58·67-s − 11.7·71-s − 0.320·73-s + 2.49·77-s − 13.5·79-s + 16.9·83-s + ⋯ |
| L(s) = 1 | + 1.86·5-s − 0.943·7-s − 0.301·11-s − 1.91·13-s − 0.376·17-s − 1.47·19-s + 1.30·23-s + 2.46·25-s + 0.897·29-s + 0.0763·31-s − 1.75·35-s − 1.73·37-s − 0.203·41-s − 0.666·43-s − 1.30·47-s − 0.110·49-s + 0.527·53-s − 0.561·55-s − 0.903·59-s + 0.673·61-s − 3.56·65-s − 0.681·67-s − 1.39·71-s − 0.0375·73-s + 0.284·77-s − 1.52·79-s + 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 - 0.425T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 + 4.36T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 - 3.84T + 53T^{2} \) |
| 59 | \( 1 + 6.93T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 0.320T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 8.16T + 97T^{2} \) |
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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
−8.421449325443680189832201670509, −7.06225316173974308289415476107, −6.73904253635269769909877553525, −6.02208164506994884559090501346, −5.11744801807506282244803743583, −4.68288829285143953322796638424, −3.09808933199363007261688948009, −2.51564585722797800135708850646, −1.69895097651857797177913546668, 0,
1.69895097651857797177913546668, 2.51564585722797800135708850646, 3.09808933199363007261688948009, 4.68288829285143953322796638424, 5.11744801807506282244803743583, 6.02208164506994884559090501346, 6.73904253635269769909877553525, 7.06225316173974308289415476107, 8.421449325443680189832201670509
