| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s − 2·13-s + 15-s + 2·17-s + 25-s − 27-s + 6·29-s + 31-s − 4·33-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s − 7·49-s − 2·51-s − 10·53-s − 4·55-s + 8·59-s + 14·61-s + 2·65-s − 8·67-s − 14·73-s − 75-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 + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s − 0.280·51-s − 1.37·53-s − 0.539·55-s + 1.04·59-s + 1.79·61-s + 0.248·65-s − 0.977·67-s − 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.48998460840191, −14.69064701054616, −14.54297601467506, −13.94687763697991, −13.08824903400449, −12.77173726332818, −12.03552220716684, −11.61653144779532, −11.45849743711882, −10.55679778651944, −9.983934957587531, −9.654418260399409, −8.845210748779746, −8.330181313582262, −7.715885522802045, −6.990357879432372, −6.626161743937994, −6.004712789830479, −5.290483926306137, −4.620448900805936, −4.172348273352471, −3.390499474272973, −2.737642655024505, −1.654869614133201, −1.013691022228280, 0,
1.013691022228280, 1.654869614133201, 2.737642655024505, 3.390499474272973, 4.172348273352471, 4.620448900805936, 5.290483926306137, 6.004712789830479, 6.626161743937994, 6.990357879432372, 7.715885522802045, 8.330181313582262, 8.845210748779746, 9.654418260399409, 9.983934957587531, 10.55679778651944, 11.45849743711882, 11.61653144779532, 12.03552220716684, 12.77173726332818, 13.08824903400449, 13.94687763697991, 14.54297601467506, 14.69064701054616, 15.48998460840191
