L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s + 2·13-s − 16-s + 6·17-s + 3·18-s − 4·19-s + 20-s + 4·23-s + 25-s − 2·26-s − 6·29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s − 2·37-s + 4·38-s − 3·40-s + 2·41-s − 4·43-s + 3·45-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.917·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.328·37-s + 0.648·38-s − 0.474·40-s + 0.312·41-s − 0.609·43-s + 0.447·45-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.32708921946808, −14.97155366152186, −14.31691262145494, −13.81865599275983, −13.45913527502187, −12.63208860957357, −12.27404668466910, −11.59415362186766, −10.95495519025865, −10.62615871276783, −10.01068453070475, −9.274148324933091, −8.948988088025391, −8.288322875556679, −7.976948089097669, −7.386971743985302, −6.649413460277163, −5.846077113448288, −5.416573407695094, −4.634962124632869, −4.028624992936016, −3.328130105971778, −2.700796572964999, −1.591735828643313, −0.8599508722406620, 0,
0.8599508722406620, 1.591735828643313, 2.700796572964999, 3.328130105971778, 4.028624992936016, 4.634962124632869, 5.416573407695094, 5.846077113448288, 6.649413460277163, 7.386971743985302, 7.976948089097669, 8.288322875556679, 8.948988088025391, 9.274148324933091, 10.01068453070475, 10.62615871276783, 10.95495519025865, 11.59415362186766, 12.27404668466910, 12.63208860957357, 13.45913527502187, 13.81865599275983, 14.31691262145494, 14.97155366152186, 15.32708921946808