L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s + 2·11-s − 2·13-s + 4·15-s + 17-s + 4·19-s − 4·21-s + 2·23-s − 25-s + 4·27-s − 2·29-s − 10·31-s − 4·33-s − 4·35-s − 10·37-s + 4·39-s + 2·41-s + 4·43-s − 2·45-s − 3·49-s − 2·51-s − 6·53-s − 4·55-s − 8·57-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.242·17-s + 0.917·19-s − 0.872·21-s + 0.417·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s − 1.79·31-s − 0.696·33-s − 0.676·35-s − 1.64·37-s + 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.473079279217919989746098035885, −8.614837538022753754311758198730, −7.55137092171531960716139749517, −7.10168065399436381239411441374, −5.87127376220204506275342473073, −5.19048124930707709395405158629, −4.34831044732353825175627159129, −3.27800062285362904687274324165, −1.51010412760179392207619869184, 0,
1.51010412760179392207619869184, 3.27800062285362904687274324165, 4.34831044732353825175627159129, 5.19048124930707709395405158629, 5.87127376220204506275342473073, 7.10168065399436381239411441374, 7.55137092171531960716139749517, 8.614837538022753754311758198730, 9.473079279217919989746098035885