| L(s) = 1 | + 16·2-s + 256·4-s + 1.79e3·5-s − 1.11e4·7-s + 4.09e3·8-s + 2.87e4·10-s + 1.84e4·11-s + 2.11e4·13-s − 1.78e5·14-s + 6.55e4·16-s − 6.60e5·17-s + 5.22e5·19-s + 4.59e5·20-s + 2.94e5·22-s − 1.04e6·23-s + 1.27e6·25-s + 3.38e5·26-s − 2.86e6·28-s − 2.60e6·29-s − 5.39e4·31-s + 1.04e6·32-s − 1.05e7·34-s − 2.00e7·35-s − 1.93e7·37-s + 8.35e6·38-s + 7.35e6·40-s + 1.29e6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.28·5-s − 1.76·7-s + 0.353·8-s + 0.908·10-s + 0.379·11-s + 0.205·13-s − 1.24·14-s + 0.250·16-s − 1.91·17-s + 0.919·19-s + 0.642·20-s + 0.268·22-s − 0.781·23-s + 0.650·25-s + 0.145·26-s − 0.880·28-s − 0.683·29-s − 0.0104·31-s + 0.176·32-s − 1.35·34-s − 2.26·35-s − 1.69·37-s + 0.649·38-s + 0.454·40-s + 0.0717·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.79e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.11e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.84e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.11e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.60e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.22e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.04e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.39e4T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.93e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.29e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.29e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.80e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.73e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.05e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.57e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.93e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.28e8T + 7.60e17T^{2} \) |
| 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
−10.56528989557654106636123511514, −9.688014556903600832389145948608, −8.948101286930504530844857400266, −6.97209598474850812382300019266, −6.33268232658512968845226271241, −5.49301710700983016079314773313, −3.96435598441207869305171230393, −2.83815430635906970432766161537, −1.76281627333280089053065074581, 0,
1.76281627333280089053065074581, 2.83815430635906970432766161537, 3.96435598441207869305171230393, 5.49301710700983016079314773313, 6.33268232658512968845226271241, 6.97209598474850812382300019266, 8.948101286930504530844857400266, 9.688014556903600832389145948608, 10.56528989557654106636123511514
