| L(s) = 1 | − 4·5-s + 7-s + 2·17-s + 2·19-s − 8·23-s + 11·25-s + 2·29-s + 4·31-s − 4·35-s + 6·37-s + 2·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s + 6·59-s − 4·61-s + 12·67-s − 14·73-s − 8·79-s + 6·83-s − 8·85-s − 10·89-s − 8·95-s − 2·97-s + 12·101-s − 12·103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s + 0.485·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.781·59-s − 0.512·61-s + 1.46·67-s − 1.63·73-s − 0.900·79-s + 0.658·83-s − 0.867·85-s − 1.05·89-s − 0.820·95-s − 0.203·97-s + 1.19·101-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.092660011067177558429540865439, −7.54801754874801697338809881828, −6.77420851319528696059978803031, −5.85675979257035438626438847546, −4.85762247801381754805120738895, −4.20793108421579356205513357787, −3.56270542712905614063510381286, −2.65043598386331574261041648988, −1.21717442621058177691884331097, 0,
1.21717442621058177691884331097, 2.65043598386331574261041648988, 3.56270542712905614063510381286, 4.20793108421579356205513357787, 4.85762247801381754805120738895, 5.85675979257035438626438847546, 6.77420851319528696059978803031, 7.54801754874801697338809881828, 8.092660011067177558429540865439
