| L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s − 3·11-s − 2·13-s + 3·14-s + 16-s + 2·17-s − 2·19-s − 20-s − 3·22-s − 4·25-s − 2·26-s + 3·28-s + 29-s + 4·31-s + 32-s + 2·34-s − 3·35-s + 8·37-s − 2·38-s − 40-s − 8·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s − 0.639·22-s − 4/5·25-s − 0.392·26-s + 0.566·28-s + 0.185·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.507·35-s + 1.31·37-s − 0.324·38-s − 0.158·40-s − 1.24·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.23863223791684, −14.87465689351900, −14.61273381895311, −13.74324062665914, −13.45077523776512, −12.88742468180475, −12.06923873309775, −11.85666260219374, −11.36781585159699, −10.68612527748144, −10.19504386279638, −9.722687517498548, −8.620943019990029, −8.302204981423853, −7.650544219091802, −7.338580561005672, −6.525431335360811, −5.762110023170315, −5.273314384767957, −4.637240837181045, −4.256620353958196, −3.407302104989351, −2.664984627931707, −2.052189150711280, −1.194301738842029, 0,
1.194301738842029, 2.052189150711280, 2.664984627931707, 3.407302104989351, 4.256620353958196, 4.637240837181045, 5.273314384767957, 5.762110023170315, 6.525431335360811, 7.338580561005672, 7.650544219091802, 8.302204981423853, 8.620943019990029, 9.722687517498548, 10.19504386279638, 10.68612527748144, 11.36781585159699, 11.85666260219374, 12.06923873309775, 12.88742468180475, 13.45077523776512, 13.74324062665914, 14.61273381895311, 14.87465689351900, 15.23863223791684
