| L(s) = 1 | + 243·3-s + 1.14e4·5-s + 8.26e3·7-s + 5.90e4·9-s − 4.15e5·11-s + 1.37e6·13-s + 2.77e6·15-s − 5.56e6·17-s + 7.16e6·19-s + 2.00e6·21-s + 4.72e7·23-s + 8.12e7·25-s + 1.43e7·27-s + 1.87e8·29-s − 1.98e8·31-s − 1.01e8·33-s + 9.42e7·35-s + 5.36e8·37-s + 3.33e8·39-s − 5.70e8·41-s − 1.07e9·43-s + 6.73e8·45-s − 2.77e9·47-s − 1.90e9·49-s − 1.35e9·51-s + 4.24e9·53-s − 4.74e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.63·5-s + 0.185·7-s + 0.333·9-s − 0.778·11-s + 1.02·13-s + 0.942·15-s − 0.951·17-s + 0.663·19-s + 0.107·21-s + 1.53·23-s + 1.66·25-s + 0.192·27-s + 1.69·29-s − 1.24·31-s − 0.449·33-s + 0.303·35-s + 1.27·37-s + 0.591·39-s − 0.769·41-s − 1.11·43-s + 0.544·45-s − 1.76·47-s − 0.965·49-s − 0.549·51-s + 1.39·53-s − 1.27·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.112800117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.112800117\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 243T \) |
| good | 5 | \( 1 - 1.14e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.26e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.15e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.37e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.56e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.16e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.72e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.87e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.98e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.70e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.07e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.77e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.24e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.85e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 6.44e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.03e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 4.66e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.25e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.55e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.17e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.72e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.57e11T + 7.15e21T^{2} \) |
| 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
−14.94003687975039569495776984778, −13.62559087944636794748690199091, −13.11173046134163389847355179893, −10.92136083175856649178733513239, −9.685134801183943987264282463023, −8.525523864705661049083597763581, −6.61169943522153195470557259100, −5.11782623323387488206287373723, −2.83904611846617850289859060829, −1.45025453044604658940685652374,
1.45025453044604658940685652374, 2.83904611846617850289859060829, 5.11782623323387488206287373723, 6.61169943522153195470557259100, 8.525523864705661049083597763581, 9.685134801183943987264282463023, 10.92136083175856649178733513239, 13.11173046134163389847355179893, 13.62559087944636794748690199091, 14.94003687975039569495776984778
