| L(s) = 1 | + 32·2-s + 419.·3-s + 1.02e3·4-s + 3.65e3·5-s + 1.34e4·6-s + 1.68e4·7-s + 3.27e4·8-s − 1.49e3·9-s + 1.16e5·10-s + 5.26e4·11-s + 4.29e5·12-s + 7.48e5·13-s + 5.37e5·14-s + 1.53e6·15-s + 1.04e6·16-s + 3.00e6·17-s − 4.77e4·18-s + 3.57e6·19-s + 3.73e6·20-s + 7.04e6·21-s + 1.68e6·22-s − 5.95e6·23-s + 1.37e7·24-s − 3.54e7·25-s + 2.39e7·26-s − 7.48e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.995·3-s + 0.5·4-s + 0.522·5-s + 0.704·6-s + 0.377·7-s + 0.353·8-s − 0.00842·9-s + 0.369·10-s + 0.0985·11-s + 0.497·12-s + 0.558·13-s + 0.267·14-s + 0.520·15-s + 0.250·16-s + 0.513·17-s − 0.00595·18-s + 0.331·19-s + 0.261·20-s + 0.376·21-s + 0.0697·22-s − 0.192·23-s + 0.352·24-s − 0.726·25-s + 0.395·26-s − 1.00·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.849992045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.849992045\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 3 | \( 1 - 419.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 3.65e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 5.26e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.48e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.00e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 3.57e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.95e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.01e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.49e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.23e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.23e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 7.78e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.33e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.02e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.16e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.89e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.28e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.52e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.48e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.26e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.32e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.71e10T + 7.15e21T^{2} \) |
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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
−16.67301681862626416936688748959, −15.03513379011746012907546888776, −14.13118803549430650143202616443, −13.08899018903895849341363141672, −11.29105313006182273846932037709, −9.398002787881878574900815282441, −7.77197779620408104624589613076, −5.70194545290817067775159412562, −3.60532769080978500785506620605, −1.94801958965765714514887987969,
1.94801958965765714514887987969, 3.60532769080978500785506620605, 5.70194545290817067775159412562, 7.77197779620408104624589613076, 9.398002787881878574900815282441, 11.29105313006182273846932037709, 13.08899018903895849341363141672, 14.13118803549430650143202616443, 15.03513379011746012907546888776, 16.67301681862626416936688748959
