| L(s) = 1 | − 3-s − 0.539·5-s − 1.70·7-s + 9-s + 11-s − 13-s + 0.539·15-s − 2.09·17-s + 6.97·19-s + 1.70·21-s + 4.78·23-s − 4.70·25-s − 27-s + 4.58·29-s + 0.199·31-s − 33-s + 0.921·35-s − 4.49·37-s + 39-s − 4.97·41-s + 1.75·43-s − 0.539·45-s − 0.156·47-s − 4.07·49-s + 2.09·51-s − 0.921·53-s − 0.539·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.241·5-s − 0.646·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.139·15-s − 0.507·17-s + 1.59·19-s + 0.372·21-s + 0.998·23-s − 0.941·25-s − 0.192·27-s + 0.852·29-s + 0.0357·31-s − 0.174·33-s + 0.155·35-s − 0.739·37-s + 0.160·39-s − 0.776·41-s + 0.267·43-s − 0.0803·45-s − 0.0228·47-s − 0.582·49-s + 0.292·51-s − 0.126·53-s − 0.0727·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197178581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197178581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 0.539T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 0.199T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 0.156T + 47T^{2} \) |
| 53 | \( 1 + 0.921T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 6.04T + 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 + 4.49T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
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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
−7.80995921966545138370704298497, −7.14682727568241222528684453749, −6.63317246131982492481675753189, −5.86149170952405278545521884595, −5.14574416103877772025771631262, −4.47483996215839261734672661909, −3.52152408584280830798695741235, −2.90513458270216527617783433690, −1.66269765356431822471279159669, −0.58491818412922617492485173584,
0.58491818412922617492485173584, 1.66269765356431822471279159669, 2.90513458270216527617783433690, 3.52152408584280830798695741235, 4.47483996215839261734672661909, 5.14574416103877772025771631262, 5.86149170952405278545521884595, 6.63317246131982492481675753189, 7.14682727568241222528684453749, 7.80995921966545138370704298497
