| L(s) = 1 | − 2.53·3-s + 247.·7-s − 236.·9-s − 290.·11-s − 119.·13-s + 1.10e3·17-s − 1.79e3·19-s − 626.·21-s − 3.05e3·23-s + 1.21e3·27-s + 2.56e3·29-s − 1.30e3·31-s + 735.·33-s − 1.50e4·37-s + 302.·39-s + 1.39e4·41-s − 1.77e3·43-s + 3.35e3·47-s + 4.43e4·49-s − 2.78e3·51-s + 2.08e4·53-s + 4.55e3·57-s − 3.16e4·59-s − 3.49e4·61-s − 5.85e4·63-s + 3.80e4·67-s + 7.74e3·69-s + ⋯ |
| L(s) = 1 | − 0.162·3-s + 1.90·7-s − 0.973·9-s − 0.723·11-s − 0.195·13-s + 0.923·17-s − 1.14·19-s − 0.310·21-s − 1.20·23-s + 0.320·27-s + 0.566·29-s − 0.243·31-s + 0.117·33-s − 1.80·37-s + 0.0318·39-s + 1.29·41-s − 0.146·43-s + 0.221·47-s + 2.64·49-s − 0.150·51-s + 1.02·53-s + 0.185·57-s − 1.18·59-s − 1.20·61-s − 1.85·63-s + 1.03·67-s + 0.195·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.53T + 243T^{2} \) |
| 7 | \( 1 - 247.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 119.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.50e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.69e4T + 8.58e9T^{2} \) |
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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
−10.32785855645634956588197382701, −8.789134433240612780568044252794, −8.183218577502833802549192974793, −7.45141427531875632694594180307, −5.90927734396216178093592916906, −5.19626106926367425525635349335, −4.21637318831574467703307275907, −2.62360057590835133565158893423, −1.54482302680741036014708047760, 0,
1.54482302680741036014708047760, 2.62360057590835133565158893423, 4.21637318831574467703307275907, 5.19626106926367425525635349335, 5.90927734396216178093592916906, 7.45141427531875632694594180307, 8.183218577502833802549192974793, 8.789134433240612780568044252794, 10.32785855645634956588197382701
