| L(s) = 1 | + 17.5·3-s − 17.9·5-s + 343·7-s − 1.87e3·9-s + 3.12e3·11-s − 1.42e4·13-s − 315.·15-s − 5.50e3·17-s + 2.90e4·19-s + 6.03e3·21-s + 3.99e3·23-s − 7.78e4·25-s − 7.14e4·27-s − 1.48e5·29-s + 2.32e5·31-s + 5.50e4·33-s − 6.15e3·35-s + 2.15e5·37-s − 2.50e5·39-s + 7.16e5·41-s − 1.57e5·43-s + 3.36e4·45-s + 1.08e6·47-s + 1.17e5·49-s − 9.68e4·51-s − 1.50e6·53-s − 5.61e4·55-s + ⋯ |
| L(s) = 1 | + 0.376·3-s − 0.0642·5-s + 0.377·7-s − 0.858·9-s + 0.708·11-s − 1.79·13-s − 0.0241·15-s − 0.271·17-s + 0.970·19-s + 0.142·21-s + 0.0684·23-s − 0.995·25-s − 0.699·27-s − 1.12·29-s + 1.40·31-s + 0.266·33-s − 0.0242·35-s + 0.701·37-s − 0.675·39-s + 1.62·41-s − 0.302·43-s + 0.0551·45-s + 1.52·47-s + 0.142·49-s − 0.102·51-s − 1.38·53-s − 0.0454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.013389732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013389732\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 17.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 17.9T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.42e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.50e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.90e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.99e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.15e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.13e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.71e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.54e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.77e6T + 8.07e13T^{2} \) |
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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
−9.608878340826532956144945091352, −9.253023775063838874700766631258, −7.964758971798018543541135494696, −7.44835633870576155781046270782, −6.17313918717515513481778629212, −5.17518600750258263121838481886, −4.16026492989531971266560595061, −2.92293892113398593119207360759, −2.05378236009486930510261385339, −0.60721144106077162679387462478,
0.60721144106077162679387462478, 2.05378236009486930510261385339, 2.92293892113398593119207360759, 4.16026492989531971266560595061, 5.17518600750258263121838481886, 6.17313918717515513481778629212, 7.44835633870576155781046270782, 7.964758971798018543541135494696, 9.253023775063838874700766631258, 9.608878340826532956144945091352
