| L(s) = 1 | + 2.05·2-s + 2.21·4-s − 0.146·5-s + 0.446·8-s − 0.300·10-s + 1.66·11-s − 0.199·13-s − 3.51·16-s − 6.27·17-s − 6.91·19-s − 0.324·20-s + 3.41·22-s − 6.18·23-s − 4.97·25-s − 0.410·26-s + 4.93·29-s − 2.51·31-s − 8.11·32-s − 12.8·34-s + 7.00·37-s − 14.2·38-s − 0.0653·40-s − 2.31·41-s + 1.88·43-s + 3.68·44-s − 12.6·46-s − 1.81·47-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.10·4-s − 0.0654·5-s + 0.157·8-s − 0.0949·10-s + 0.501·11-s − 0.0554·13-s − 0.879·16-s − 1.52·17-s − 1.58·19-s − 0.0725·20-s + 0.728·22-s − 1.28·23-s − 0.995·25-s − 0.0805·26-s + 0.916·29-s − 0.452·31-s − 1.43·32-s − 2.20·34-s + 1.15·37-s − 2.30·38-s − 0.0103·40-s − 0.361·41-s + 0.287·43-s + 0.556·44-s − 1.87·46-s − 0.264·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 + 0.146T + 5T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 + 0.199T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 4.93T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 - 5.34T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 0.678T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 - 1.55T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−8.051646146247642738722547380615, −7.00612646158103542469896316031, −6.28455053466980276366069426670, −5.99030322559360517209687260492, −4.84458148715975042450757971614, −4.23493706201160287159948748323, −3.80219887562824720413602279060, −2.58183711419704567511085454097, −1.94822330199233565496904883250, 0,
1.94822330199233565496904883250, 2.58183711419704567511085454097, 3.80219887562824720413602279060, 4.23493706201160287159948748323, 4.84458148715975042450757971614, 5.99030322559360517209687260492, 6.28455053466980276366069426670, 7.00612646158103542469896316031, 8.051646146247642738722547380615
