| L(s) = 1 | + 0.561·5-s + 0.438·7-s − 5.56·11-s + 13-s − 4·17-s + 5.12·19-s + 6.68·23-s − 4.68·25-s − 2.68·29-s − 6.68·31-s + 0.246·35-s + 6.68·37-s − 12.2·41-s − 1.87·43-s + 3.68·47-s − 6.80·49-s + 1.56·53-s − 3.12·55-s − 2.12·59-s − 14.5·61-s + 0.561·65-s + 10.2·67-s − 7.68·71-s − 12·73-s − 2.43·77-s + 8·79-s − 17.2·83-s + ⋯ |
| L(s) = 1 | + 0.251·5-s + 0.165·7-s − 1.67·11-s + 0.277·13-s − 0.970·17-s + 1.17·19-s + 1.39·23-s − 0.936·25-s − 0.498·29-s − 1.20·31-s + 0.0416·35-s + 1.09·37-s − 1.91·41-s − 0.286·43-s + 0.537·47-s − 0.972·49-s + 0.214·53-s − 0.421·55-s − 0.276·59-s − 1.86·61-s + 0.0696·65-s + 1.25·67-s − 0.912·71-s − 1.40·73-s − 0.277·77-s + 0.900·79-s − 1.89·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.398832495251478827232732214135, −7.61535106580998324094938792885, −7.07624949049831260516644937760, −5.98492608229808507628673856736, −5.29379124278750908746513348198, −4.67902431801573317527577020400, −3.44689735577252670140106788892, −2.64506894971593902596127459427, −1.59496940392044983676594417019, 0,
1.59496940392044983676594417019, 2.64506894971593902596127459427, 3.44689735577252670140106788892, 4.67902431801573317527577020400, 5.29379124278750908746513348198, 5.98492608229808507628673856736, 7.07624949049831260516644937760, 7.61535106580998324094938792885, 8.398832495251478827232732214135
