| L(s) = 1 | + 27·3-s − 1.00e3·7-s + 729·9-s + 6.31e3·11-s − 1.19e4·13-s + 288.·17-s + 2.80e4·19-s − 2.70e4·21-s − 2.82e4·23-s + 1.96e4·27-s + 9.76e4·29-s − 2.90e5·31-s + 1.70e5·33-s + 2.47e4·37-s − 3.21e5·39-s + 6.88e5·41-s − 3.96e5·43-s + 9.20e5·47-s + 1.82e5·49-s + 7.78e3·51-s + 1.64e6·53-s + 7.56e5·57-s + 2.34e6·59-s − 1.28e5·61-s − 7.31e5·63-s + 2.69e6·67-s − 7.63e5·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.10·7-s + 0.333·9-s + 1.42·11-s − 1.50·13-s + 0.0142·17-s + 0.936·19-s − 0.637·21-s − 0.484·23-s + 0.192·27-s + 0.743·29-s − 1.75·31-s + 0.825·33-s + 0.0802·37-s − 0.867·39-s + 1.56·41-s − 0.759·43-s + 1.29·47-s + 0.221·49-s + 0.00822·51-s + 1.52·53-s + 0.540·57-s + 1.48·59-s − 0.0725·61-s − 0.368·63-s + 1.09·67-s − 0.279·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.290577647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290577647\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.00e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.19e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 288.T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.76e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.90e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.47e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.20e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.64e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.28e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.11e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.94e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.90e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18674923836694175504725612596, −9.525850825015681818755161602477, −8.890985069529165998983815323077, −7.46659143816586622491983101490, −6.83742405591916982976147692949, −5.64378606945454018887428389007, −4.22281915224765162409211665291, −3.29126618792892060418968051515, −2.17762608584706220069344512215, −0.71079289107837677449143503591,
0.71079289107837677449143503591, 2.17762608584706220069344512215, 3.29126618792892060418968051515, 4.22281915224765162409211665291, 5.64378606945454018887428389007, 6.83742405591916982976147692949, 7.46659143816586622491983101490, 8.890985069529165998983815323077, 9.525850825015681818755161602477, 10.18674923836694175504725612596
