| L(s) = 1 | − 1.56·2-s + 3-s + 0.446·4-s + 1.37·5-s − 1.56·6-s − 0.139·7-s + 2.43·8-s + 9-s − 2.14·10-s − 0.497·11-s + 0.446·12-s + 2.04·13-s + 0.218·14-s + 1.37·15-s − 4.69·16-s − 5.28·17-s − 1.56·18-s − 2.86·19-s + 0.612·20-s − 0.139·21-s + 0.777·22-s + 2.32·23-s + 2.43·24-s − 3.11·25-s − 3.20·26-s + 27-s − 0.0622·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.223·4-s + 0.613·5-s − 0.638·6-s − 0.0527·7-s + 0.859·8-s + 0.333·9-s − 0.678·10-s − 0.149·11-s + 0.128·12-s + 0.568·13-s + 0.0583·14-s + 0.354·15-s − 1.17·16-s − 1.28·17-s − 0.368·18-s − 0.658·19-s + 0.136·20-s − 0.0304·21-s + 0.165·22-s + 0.484·23-s + 0.496·24-s − 0.623·25-s − 0.628·26-s + 0.192·27-s − 0.0117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 0.139T + 7T^{2} \) |
| 11 | \( 1 + 0.497T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.64T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 - 8.46T + 97T^{2} \) |
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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
−8.514752132692552960241559094556, −8.006023125717003931958130055630, −6.96104185854732040303822325572, −6.46453907923888628242014404803, −5.27586577519272883724010388534, −4.42548715653310021508403806544, −3.46692076615910286122758085885, −2.19245045346378353815098595572, −1.56251653880309234025377893516, 0,
1.56251653880309234025377893516, 2.19245045346378353815098595572, 3.46692076615910286122758085885, 4.42548715653310021508403806544, 5.27586577519272883724010388534, 6.46453907923888628242014404803, 6.96104185854732040303822325572, 8.006023125717003931958130055630, 8.514752132692552960241559094556
