| L(s) = 1 | − 1.34·2-s − 3-s − 0.185·4-s − 2.41·5-s + 1.34·6-s − 7-s + 2.94·8-s + 9-s + 3.24·10-s + 2.16·11-s + 0.185·12-s + 6.32·13-s + 1.34·14-s + 2.41·15-s − 3.59·16-s + 3.75·17-s − 1.34·18-s − 3.21·19-s + 0.447·20-s + 21-s − 2.91·22-s − 7.17·23-s − 2.94·24-s + 0.816·25-s − 8.51·26-s − 27-s + 0.185·28-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.577·3-s − 0.0928·4-s − 1.07·5-s + 0.549·6-s − 0.377·7-s + 1.04·8-s + 0.333·9-s + 1.02·10-s + 0.652·11-s + 0.0536·12-s + 1.75·13-s + 0.359·14-s + 0.622·15-s − 0.898·16-s + 0.911·17-s − 0.317·18-s − 0.737·19-s + 0.100·20-s + 0.218·21-s − 0.621·22-s − 1.49·23-s − 0.600·24-s + 0.163·25-s − 1.66·26-s − 0.192·27-s + 0.0351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 + 3.21T + 19T^{2} \) |
| 23 | \( 1 + 7.17T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 - 0.383T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.964T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−7.69413831658603361539160801460, −6.91010273251771718527797605864, −6.19117197610014275406495531651, −5.59775297880305388910269765038, −4.33255388455631530351127245618, −4.05793260788843255555800992862, −3.32632676154045089337463840685, −1.76002287367290591541964329304, −0.943654328616491748618116940093, 0,
0.943654328616491748618116940093, 1.76002287367290591541964329304, 3.32632676154045089337463840685, 4.05793260788843255555800992862, 4.33255388455631530351127245618, 5.59775297880305388910269765038, 6.19117197610014275406495531651, 6.91010273251771718527797605864, 7.69413831658603361539160801460
