| L(s) = 1 | + 1.36·2-s − 3-s − 0.138·4-s + 3.32·5-s − 1.36·6-s + 7-s − 2.91·8-s + 9-s + 4.53·10-s − 1.11·11-s + 0.138·12-s + 1.36·14-s − 3.32·15-s − 3.70·16-s + 0.665·17-s + 1.36·18-s − 3.22·19-s − 0.458·20-s − 21-s − 1.51·22-s + 5.07·23-s + 2.91·24-s + 6.02·25-s − 27-s − 0.138·28-s − 0.615·29-s − 4.53·30-s + ⋯ |
| L(s) = 1 | + 0.964·2-s − 0.577·3-s − 0.0690·4-s + 1.48·5-s − 0.557·6-s + 0.377·7-s − 1.03·8-s + 0.333·9-s + 1.43·10-s − 0.335·11-s + 0.0398·12-s + 0.364·14-s − 0.857·15-s − 0.926·16-s + 0.161·17-s + 0.321·18-s − 0.740·19-s − 0.102·20-s − 0.218·21-s − 0.324·22-s + 1.05·23-s + 0.595·24-s + 1.20·25-s − 0.192·27-s − 0.0261·28-s − 0.114·29-s − 0.827·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.022102558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.022102558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 17 | \( 1 - 0.665T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 + 0.615T + 29T^{2} \) |
| 31 | \( 1 - 8.84T + 31T^{2} \) |
| 37 | \( 1 - 3.79T + 37T^{2} \) |
| 41 | \( 1 + 0.916T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + 5.48T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−8.801226511198851199420272635064, −7.69302309242356602190652180037, −6.67139945450194036006451567426, −6.01424946158033734097234568277, −5.59649442206019795917759184506, −4.80728131364311689593094432119, −4.27620508711901499285566878717, −2.97130120239577569173739295520, −2.22750659303366311974446129883, −0.939414295099444037241247428575,
0.939414295099444037241247428575, 2.22750659303366311974446129883, 2.97130120239577569173739295520, 4.27620508711901499285566878717, 4.80728131364311689593094432119, 5.59649442206019795917759184506, 6.01424946158033734097234568277, 6.67139945450194036006451567426, 7.69302309242356602190652180037, 8.801226511198851199420272635064
