L(s) = 1 | − 9.38·2-s − 9·3-s + 56.1·4-s − 71.7·5-s + 84.5·6-s − 226.·8-s + 81·9-s + 673.·10-s − 560.·11-s − 505.·12-s + 533.·13-s + 645.·15-s + 333.·16-s + 1.00e3·17-s − 760.·18-s + 1.36e3·19-s − 4.02e3·20-s + 5.26e3·22-s + 3.22e3·23-s + 2.04e3·24-s + 2.02e3·25-s − 5.00e3·26-s − 729·27-s − 753.·29-s − 6.06e3·30-s + 8.20e3·31-s + 4.12e3·32-s + ⋯ |
L(s) = 1 | − 1.65·2-s − 0.577·3-s + 1.75·4-s − 1.28·5-s + 0.958·6-s − 1.25·8-s + 0.333·9-s + 2.12·10-s − 1.39·11-s − 1.01·12-s + 0.875·13-s + 0.740·15-s + 0.325·16-s + 0.844·17-s − 0.553·18-s + 0.869·19-s − 2.25·20-s + 2.31·22-s + 1.27·23-s + 0.723·24-s + 0.646·25-s − 1.45·26-s − 0.192·27-s − 0.166·29-s − 1.22·30-s + 1.53·31-s + 0.712·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 9.38T + 32T^{2} \) |
| 5 | \( 1 + 71.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 560.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 533.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 753.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 245.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.75e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 954.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.29e4T + 8.58e9T^{2} \) |
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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
−11.28552048779940648937517277410, −10.62756941014156541219263681440, −9.622740497705371199578976456630, −8.224738214841194984616247590072, −7.80523390741995263881636308002, −6.69067382447224610595070230210, −5.03780920621466594659905877487, −3.16910514019611927254661152818, −1.10574336901116235880305312221, 0,
1.10574336901116235880305312221, 3.16910514019611927254661152818, 5.03780920621466594659905877487, 6.69067382447224610595070230210, 7.80523390741995263881636308002, 8.224738214841194984616247590072, 9.622740497705371199578976456630, 10.62756941014156541219263681440, 11.28552048779940648937517277410