| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s − 6·17-s − 4·19-s + 2·20-s + 4·22-s − 25-s − 2·26-s − 28-s + 2·29-s + 5·32-s − 6·34-s − 2·35-s − 6·37-s − 4·38-s + 6·40-s − 2·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s + 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.986·37-s − 0.648·38-s + 0.948·40-s − 0.312·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8689266066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8689266066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.98965123574319, −14.50837438527650, −13.85271396623249, −13.63960656248035, −12.82498686361913, −12.35653456813616, −11.96435588166499, −11.47552855150514, −10.91932215965011, −10.31238839231699, −9.417369203897864, −9.119807297535601, −8.436082917442770, −8.163268786660262, −7.176799994749514, −6.777124866432225, −6.145251023360417, −5.452364010217351, −4.688348636630106, −4.221515233408947, −4.001117952713644, −3.157705580799969, −2.372253377790392, −1.487830945243176, −0.3147800536775853,
0.3147800536775853, 1.487830945243176, 2.372253377790392, 3.157705580799969, 4.001117952713644, 4.221515233408947, 4.688348636630106, 5.452364010217351, 6.145251023360417, 6.777124866432225, 7.176799994749514, 8.163268786660262, 8.436082917442770, 9.119807297535601, 9.417369203897864, 10.31238839231699, 10.91932215965011, 11.47552855150514, 11.96435588166499, 12.35653456813616, 12.82498686361913, 13.63960656248035, 13.85271396623249, 14.50837438527650, 14.98965123574319
