| L(s) = 1 | + 3-s + 7-s + 9-s − 2·13-s − 8·17-s − 19-s + 21-s − 4·23-s − 5·25-s + 27-s − 2·37-s − 2·39-s + 2·41-s − 4·43-s − 6·47-s + 49-s − 8·51-s − 12·53-s − 57-s − 4·59-s − 2·61-s + 63-s + 4·67-s − 4·69-s + 6·71-s + 6·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.94·17-s − 0.229·19-s + 0.218·21-s − 0.834·23-s − 25-s + 0.192·27-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s − 1.12·51-s − 1.64·53-s − 0.132·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s + 0.712·71-s + 0.702·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.208329763915009155875693274441, −7.75381208178629701986032179519, −6.78432954817853412246616257104, −6.20105211219491075074899961625, −5.02692945253514422792587878059, −4.40011931011779862680526321493, −3.55875172199588887914229143879, −2.39741119584776535278731996108, −1.79152280881413347668477476302, 0,
1.79152280881413347668477476302, 2.39741119584776535278731996108, 3.55875172199588887914229143879, 4.40011931011779862680526321493, 5.02692945253514422792587878059, 6.20105211219491075074899961625, 6.78432954817853412246616257104, 7.75381208178629701986032179519, 8.208329763915009155875693274441
