| L(s) = 1 | − 26.4·3-s − 35.9·7-s + 459.·9-s + 436.·11-s − 16.3·13-s + 1.32e3·17-s + 1.14e3·19-s + 951.·21-s + 940.·23-s − 5.72e3·27-s + 8.38e3·29-s + 701.·31-s − 1.15e4·33-s + 5.33e3·37-s + 432.·39-s + 9.37e3·41-s − 1.73e4·43-s − 1.64e4·47-s − 1.55e4·49-s − 3.50e4·51-s + 3.82e4·53-s − 3.02e4·57-s + 4.98e4·59-s − 3.29e4·61-s − 1.64e4·63-s − 5.70e4·67-s − 2.49e4·69-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.277·7-s + 1.88·9-s + 1.08·11-s − 0.0268·13-s + 1.11·17-s + 0.725·19-s + 0.470·21-s + 0.370·23-s − 1.51·27-s + 1.85·29-s + 0.131·31-s − 1.84·33-s + 0.640·37-s + 0.0455·39-s + 0.871·41-s − 1.43·43-s − 1.08·47-s − 0.923·49-s − 1.88·51-s + 1.87·53-s − 1.23·57-s + 1.86·59-s − 1.13·61-s − 0.523·63-s − 1.55·67-s − 0.629·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.387457690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387457690\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 26.4T + 243T^{2} \) |
| 7 | \( 1 + 35.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 436.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 16.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 940.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 701.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.73e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.64e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.01e4T + 8.58e9T^{2} \) |
| 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
−9.924781592745175563131984881044, −8.759771038786402657942510893087, −7.51168158520935486319636438137, −6.65569832114587952586288287473, −6.06314766189388613387424502317, −5.18079267203606189855397353276, −4.35713046559113290702811277329, −3.16640435690119182871895269909, −1.35332348861794018666298000155, −0.67705556849181180200237450365,
0.67705556849181180200237450365, 1.35332348861794018666298000155, 3.16640435690119182871895269909, 4.35713046559113290702811277329, 5.18079267203606189855397353276, 6.06314766189388613387424502317, 6.65569832114587952586288287473, 7.51168158520935486319636438137, 8.759771038786402657942510893087, 9.924781592745175563131984881044
