| L(s) = 1 | − 26.2·3-s − 23.9·7-s + 447.·9-s + 704.·11-s − 1.03e3·13-s − 217.·17-s − 1.29e3·19-s + 628.·21-s − 2.80e3·23-s − 5.36e3·27-s + 4.90e3·29-s + 415.·31-s − 1.84e4·33-s + 6.45e3·37-s + 2.72e4·39-s − 1.64e4·41-s + 1.37e4·43-s + 1.29e4·47-s − 1.62e4·49-s + 5.71e3·51-s + 5.23e3·53-s + 3.38e4·57-s + 2.40e4·59-s − 1.03e3·61-s − 1.07e4·63-s + 4.79e4·67-s + 7.37e4·69-s + ⋯ |
| L(s) = 1 | − 1.68·3-s − 0.184·7-s + 1.84·9-s + 1.75·11-s − 1.70·13-s − 0.182·17-s − 0.819·19-s + 0.311·21-s − 1.10·23-s − 1.41·27-s + 1.08·29-s + 0.0776·31-s − 2.95·33-s + 0.774·37-s + 2.86·39-s − 1.52·41-s + 1.13·43-s + 0.854·47-s − 0.965·49-s + 0.307·51-s + 0.256·53-s + 1.38·57-s + 0.899·59-s − 0.0356·61-s − 0.339·63-s + 1.30·67-s + 1.86·69-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 26.2T + 243T^{2} \) |
| 7 | \( 1 + 23.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 704.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 217.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.29e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 415.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.64e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.23e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.03e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.41e4T + 8.58e9T^{2} \) |
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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
−9.446811450757839828578294368998, −8.145167999059180848135402522532, −6.81129288394898467616476215996, −6.61816820023253336436080276217, −5.60100319865867240098843633433, −4.66185524095237171021191708034, −3.96928585101457014901996801029, −2.20876803157991871289703027301, −0.959584701177068954682738527428, 0,
0.959584701177068954682738527428, 2.20876803157991871289703027301, 3.96928585101457014901996801029, 4.66185524095237171021191708034, 5.60100319865867240098843633433, 6.61816820023253336436080276217, 6.81129288394898467616476215996, 8.145167999059180848135402522532, 9.446811450757839828578294368998
