| L(s) = 1 | − 2-s − 0.539·3-s + 4-s + 0.539·6-s − 0.709·7-s − 8-s − 2.70·9-s + 4.51·11-s − 0.539·12-s + 0.709·14-s + 16-s + 3.17·17-s + 2.70·18-s + 0.120·19-s + 0.382·21-s − 4.51·22-s + 4.97·23-s + 0.539·24-s + 3.07·27-s − 0.709·28-s − 7.26·29-s − 9.66·31-s − 32-s − 2.43·33-s − 3.17·34-s − 2.70·36-s + 6.95·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.311·3-s + 0.5·4-s + 0.220·6-s − 0.268·7-s − 0.353·8-s − 0.903·9-s + 1.35·11-s − 0.155·12-s + 0.189·14-s + 0.250·16-s + 0.768·17-s + 0.638·18-s + 0.0276·19-s + 0.0834·21-s − 0.961·22-s + 1.03·23-s + 0.110·24-s + 0.592·27-s − 0.134·28-s − 1.34·29-s − 1.73·31-s − 0.176·32-s − 0.423·33-s − 0.543·34-s − 0.451·36-s + 1.14·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143169379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143169379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 7 | \( 1 + 0.709T + 7T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 - 0.120T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 - 6.95T + 37T^{2} \) |
| 41 | \( 1 + 0.447T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 0.355T + 83T^{2} \) |
| 89 | \( 1 + 3.63T + 89T^{2} \) |
| 97 | \( 1 - 7.90T + 97T^{2} \) |
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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
−7.75509761766054237261758705890, −7.11857661384513822633912849641, −6.53568778001151075341615522120, −5.71075163220453676624978367068, −5.35605105176659356675291823823, −4.07004804331544984966465980337, −3.45628063324341093371679032317, −2.57606842189634129244226578282, −1.53393777951295792797140442393, −0.61738396197743771127397873162,
0.61738396197743771127397873162, 1.53393777951295792797140442393, 2.57606842189634129244226578282, 3.45628063324341093371679032317, 4.07004804331544984966465980337, 5.35605105176659356675291823823, 5.71075163220453676624978367068, 6.53568778001151075341615522120, 7.11857661384513822633912849641, 7.75509761766054237261758705890
