| L(s) = 1 | − 1.73·3-s − 3.07·5-s + 4.60·7-s + 0.0139·9-s + 3.25·11-s − 3.13·13-s + 5.33·15-s + 1.98·17-s − 3.15·19-s − 7.99·21-s + 8.21·23-s + 4.44·25-s + 5.18·27-s − 1.32·29-s − 0.155·31-s − 5.64·33-s − 14.1·35-s − 0.122·37-s + 5.43·39-s − 6.73·41-s + 10.2·43-s − 0.0429·45-s − 47-s + 14.2·49-s − 3.44·51-s + 9.44·53-s − 9.99·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 1.37·5-s + 1.74·7-s + 0.00465·9-s + 0.980·11-s − 0.868·13-s + 1.37·15-s + 0.481·17-s − 0.724·19-s − 1.74·21-s + 1.71·23-s + 0.889·25-s + 0.997·27-s − 0.245·29-s − 0.0278·31-s − 0.982·33-s − 2.39·35-s − 0.0201·37-s + 0.870·39-s − 1.05·41-s + 1.56·43-s − 0.00639·45-s − 0.145·47-s + 2.03·49-s − 0.482·51-s + 1.29·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138339853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138339853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 - 1.98T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 + 0.155T + 31T^{2} \) |
| 37 | \( 1 + 0.122T + 37T^{2} \) |
| 41 | \( 1 + 6.73T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 53 | \( 1 - 9.44T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 0.165T + 61T^{2} \) |
| 67 | \( 1 + 0.709T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 2.71T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 2.82T + 97T^{2} \) |
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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
−7.937635216101452587350079390159, −7.40207785746490159491971807803, −6.80589757290534584664494527378, −5.82238775133438051168671575466, −5.02019781262580184634709997343, −4.60935249871407366319705996659, −3.93295813967137021958065701065, −2.83836087695360298238661053959, −1.55034219801182379806008611627, −0.62885356895744144886002033301,
0.62885356895744144886002033301, 1.55034219801182379806008611627, 2.83836087695360298238661053959, 3.93295813967137021958065701065, 4.60935249871407366319705996659, 5.02019781262580184634709997343, 5.82238775133438051168671575466, 6.80589757290534584664494527378, 7.40207785746490159491971807803, 7.937635216101452587350079390159
