| L(s) = 1 | + 4.24·3-s − 65.0·5-s − 225·9-s − 274·11-s − 746.·13-s − 276·15-s − 171.·17-s + 575.·19-s − 2.75e3·23-s + 1.10e3·25-s − 1.98e3·27-s − 3.98e3·29-s − 7.42e3·31-s − 1.16e3·33-s + 1.24e4·37-s − 3.16e3·39-s − 1.12e4·41-s − 1.70e4·43-s + 1.46e4·45-s + 1.71e4·47-s − 725.·51-s + 3.35e4·53-s + 1.78e4·55-s + 2.44e3·57-s − 5.27e4·59-s − 2.87e4·61-s + 4.85e4·65-s + ⋯ |
| L(s) = 1 | + 0.272·3-s − 1.16·5-s − 0.925·9-s − 0.682·11-s − 1.22·13-s − 0.316·15-s − 0.143·17-s + 0.365·19-s − 1.08·23-s + 0.354·25-s − 0.524·27-s − 0.879·29-s − 1.38·31-s − 0.185·33-s + 1.49·37-s − 0.333·39-s − 1.04·41-s − 1.40·43-s + 1.07·45-s + 1.13·47-s − 0.0390·51-s + 1.64·53-s + 0.794·55-s + 0.0995·57-s − 1.97·59-s − 0.990·61-s + 1.42·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3065543850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3065543850\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4.24T + 243T^{2} \) |
| 5 | \( 1 + 65.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 274T + 1.61e5T^{2} \) |
| 13 | \( 1 + 746.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 171.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 575.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.87e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.54e4T + 8.58e9T^{2} \) |
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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
−9.470992962471521965258669858449, −8.565486551326832440270991137900, −7.72237893339769620344335054529, −7.34964237013917137550833884216, −5.93132861176516595887146190554, −5.04008038984725728382232927432, −3.98901872633266974571688997546, −3.07401814334093639232249783631, −2.08162691289654556715088404573, −0.23267906574130013221452631315,
0.23267906574130013221452631315, 2.08162691289654556715088404573, 3.07401814334093639232249783631, 3.98901872633266974571688997546, 5.04008038984725728382232927432, 5.93132861176516595887146190554, 7.34964237013917137550833884216, 7.72237893339769620344335054529, 8.565486551326832440270991137900, 9.470992962471521965258669858449
