| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s − 2·13-s + 16-s − 17-s − 18-s + 4·23-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·29-s − 32-s + 34-s + 36-s + 8·37-s + 4·39-s − 2·41-s − 4·46-s − 2·48-s + 5·50-s + 2·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.834·23-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.742·29-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.589·46-s − 0.288·48-s + 0.707·50-s + 0.280·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.27267309524393, −12.62707314120141, −12.15355940263847, −11.83561795930972, −11.33201677816529, −10.92803631364085, −10.63575519737163, −9.988665952418540, −9.562457747896264, −9.170549996600480, −8.562673548005999, −8.051761552054857, −7.421402893243347, −7.183994594903921, −6.475741000908711, −6.093551841942840, −5.617482219649494, −5.152893518911853, −4.502173439453270, −4.080919429454946, −3.126556444807816, −2.748509466126829, −1.932495588187494, −1.359126158984857, −0.5779667434998722, 0,
0.5779667434998722, 1.359126158984857, 1.932495588187494, 2.748509466126829, 3.126556444807816, 4.080919429454946, 4.502173439453270, 5.152893518911853, 5.617482219649494, 6.093551841942840, 6.475741000908711, 7.183994594903921, 7.421402893243347, 8.051761552054857, 8.562673548005999, 9.170549996600480, 9.562457747896264, 9.988665952418540, 10.63575519737163, 10.92803631364085, 11.33201677816529, 11.83561795930972, 12.15355940263847, 12.62707314120141, 13.27267309524393
