| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s − 8·19-s + 25-s + 27-s − 2·29-s + 4·33-s − 6·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s − 8·47-s + 2·51-s − 6·53-s − 4·55-s − 8·57-s + 2·61-s − 2·65-s + 4·67-s + 4·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 1.83·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.539·55-s − 1.05·57-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.474·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.63114206195927, −15.03485215911274, −14.71230976919032, −14.19794423206639, −13.61726766535513, −13.04965262176222, −12.33163214970432, −12.18246830430157, −11.13217421543722, −11.03133840293512, −10.20045894592432, −9.558338512870928, −9.050005206980134, −8.393393034993039, −8.166921684081035, −7.314682928549222, −6.613311647132142, −6.356991405496237, −5.417845886370063, −4.629673052064090, −3.884614914240310, −3.681031084678159, −2.750536768277644, −1.871258497428712, −1.227165305958462, 0,
1.227165305958462, 1.871258497428712, 2.750536768277644, 3.681031084678159, 3.884614914240310, 4.629673052064090, 5.417845886370063, 6.356991405496237, 6.613311647132142, 7.314682928549222, 8.166921684081035, 8.393393034993039, 9.050005206980134, 9.558338512870928, 10.20045894592432, 11.03133840293512, 11.13217421543722, 12.18246830430157, 12.33163214970432, 13.04965262176222, 13.61726766535513, 14.19794423206639, 14.71230976919032, 15.03485215911274, 15.63114206195927
