| L(s) = 1 | + 122.·7-s + 100·11-s + 734.·13-s + 979.·17-s − 2.24e3·19-s + 3.41e3·23-s + 7.85e3·29-s − 2.14e3·31-s − 1.04e4·37-s + 7.41e3·41-s + 1.77e4·43-s − 9.43e3·47-s − 1.80e3·49-s + 2.42e4·53-s − 2.59e4·59-s − 3.05e3·61-s − 5.87e4·67-s − 3.76e4·71-s + 2.40e4·73-s + 1.22e4·77-s + 7.97e4·79-s − 1.62e4·83-s + 826·89-s + 9.00e4·91-s + 3.75e4·97-s + 1.43e5·101-s − 1.11e5·103-s + ⋯ |
| L(s) = 1 | + 0.944·7-s + 0.249·11-s + 1.20·13-s + 0.822·17-s − 1.42·19-s + 1.34·23-s + 1.73·29-s − 0.400·31-s − 1.24·37-s + 0.688·41-s + 1.46·43-s − 0.622·47-s − 0.107·49-s + 1.18·53-s − 0.971·59-s − 0.105·61-s − 1.59·67-s − 0.885·71-s + 0.527·73-s + 0.235·77-s + 1.43·79-s − 0.259·83-s + 0.0110·89-s + 1.13·91-s + 0.405·97-s + 1.40·101-s − 1.03·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.071046222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.071046222\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 122.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 100T + 1.61e5T^{2} \) |
| 13 | \( 1 - 734.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 979.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 826T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.75e4T + 8.58e9T^{2} \) |
| 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
−9.126700451434362867121057890701, −8.574257489494151579566384356348, −7.80431377301478177944861334491, −6.76865468579964651522279690780, −5.93141653463531801195932272034, −4.91849411603218472858510013137, −4.07858779606612096654399962919, −2.97107429671414831288879183822, −1.68627440622237881937487468594, −0.839836996627529647435414622023,
0.839836996627529647435414622023, 1.68627440622237881937487468594, 2.97107429671414831288879183822, 4.07858779606612096654399962919, 4.91849411603218472858510013137, 5.93141653463531801195932272034, 6.76865468579964651522279690780, 7.80431377301478177944861334491, 8.574257489494151579566384356348, 9.126700451434362867121057890701
