| L(s) = 1 | − 3-s − 2·9-s + 11-s + 6·13-s + 7·17-s + 19-s + 8·23-s + 5·27-s + 6·29-s − 4·31-s − 33-s − 8·37-s − 6·39-s + 5·41-s − 6·47-s − 7·51-s − 4·53-s − 57-s − 4·59-s + 6·61-s + 5·67-s − 8·69-s + 14·71-s − 15·73-s + 14·79-s + 81-s + 83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s + 1.66·13-s + 1.69·17-s + 0.229·19-s + 1.66·23-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s − 1.31·37-s − 0.960·39-s + 0.780·41-s − 0.875·47-s − 0.980·51-s − 0.549·53-s − 0.132·57-s − 0.520·59-s + 0.768·61-s + 0.610·67-s − 0.963·69-s + 1.66·71-s − 1.75·73-s + 1.57·79-s + 1/9·81-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.763934765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.763934765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.02501736385874, −13.59116190015872, −12.94182846651486, −12.47284193950837, −12.00007564670355, −11.49290304537471, −10.99166851090120, −10.70806378259548, −10.12761365183268, −9.382435488848826, −9.019374319689091, −8.319798715581141, −8.119399116295932, −7.248239148012777, −6.732247289790611, −6.216460200460616, −5.687497524107228, −5.240385749706920, −4.740672203009844, −3.811009568363151, −3.277186639832110, −2.986998368759755, −1.842274971714669, −1.084524486354323, −0.6872041914324070,
0.6872041914324070, 1.084524486354323, 1.842274971714669, 2.986998368759755, 3.277186639832110, 3.811009568363151, 4.740672203009844, 5.240385749706920, 5.687497524107228, 6.216460200460616, 6.732247289790611, 7.248239148012777, 8.119399116295932, 8.319798715581141, 9.019374319689091, 9.382435488848826, 10.12761365183268, 10.70806378259548, 10.99166851090120, 11.49290304537471, 12.00007564670355, 12.47284193950837, 12.94182846651486, 13.59116190015872, 14.02501736385874
