| L(s) = 1 | + 1.47·2-s − 3-s + 0.172·4-s − 3.62·5-s − 1.47·6-s − 2.69·8-s + 9-s − 5.34·10-s + 1.81·11-s − 0.172·12-s − 3.13·13-s + 3.62·15-s − 4.31·16-s − 3.47·17-s + 1.47·18-s − 1.56·19-s − 0.624·20-s + 2.66·22-s − 23-s + 2.69·24-s + 8.12·25-s − 4.61·26-s − 27-s − 4.13·29-s + 5.34·30-s + 1.63·31-s − 0.972·32-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.577·3-s + 0.0862·4-s − 1.62·5-s − 0.601·6-s − 0.952·8-s + 0.333·9-s − 1.68·10-s + 0.546·11-s − 0.0497·12-s − 0.869·13-s + 0.935·15-s − 1.07·16-s − 0.841·17-s + 0.347·18-s − 0.358·19-s − 0.139·20-s + 0.569·22-s − 0.208·23-s + 0.549·24-s + 1.62·25-s − 0.905·26-s − 0.192·27-s − 0.767·29-s + 0.975·30-s + 0.294·31-s − 0.171·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8435740694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8435740694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 - 1.63T + 31T^{2} \) |
| 37 | \( 1 - 0.807T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 7.98T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 5.16T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.465722059033690229007903930442, −7.81414872884215936676610397078, −6.79620908121856327845817796855, −6.49167621787511875001610571152, −5.18248867257899286009683912908, −4.82424083586045779998198006123, −3.90884627876514267615177362105, −3.59413079931326528601796062410, −2.29836924991142288865794297858, −0.45844616738825080043323794578,
0.45844616738825080043323794578, 2.29836924991142288865794297858, 3.59413079931326528601796062410, 3.90884627876514267615177362105, 4.82424083586045779998198006123, 5.18248867257899286009683912908, 6.49167621787511875001610571152, 6.79620908121856327845817796855, 7.81414872884215936676610397078, 8.465722059033690229007903930442
