| L(s) = 1 | + 1.16·2-s − 0.645·4-s − 1.64·5-s + 1.16·7-s − 3.07·8-s − 1.91·10-s + 4.24·13-s + 1.35·14-s − 2.29·16-s + 3.07·17-s − 6.57·19-s + 1.06·20-s + 1.35·23-s − 2.29·25-s + 4.93·26-s − 0.751·28-s − 1.16·29-s − 3.64·31-s + 3.49·32-s + 3.58·34-s − 1.91·35-s − 3.35·37-s − 7.64·38-s + 5.06·40-s + 9.64·41-s + 10.0·43-s + 1.57·46-s + ⋯ |
| L(s) = 1 | + 0.822·2-s − 0.322·4-s − 0.736·5-s + 0.439·7-s − 1.08·8-s − 0.605·10-s + 1.17·13-s + 0.361·14-s − 0.572·16-s + 0.746·17-s − 1.50·19-s + 0.237·20-s + 0.282·23-s − 0.458·25-s + 0.968·26-s − 0.142·28-s − 0.216·29-s − 0.654·31-s + 0.617·32-s + 0.614·34-s − 0.323·35-s − 0.551·37-s − 1.24·38-s + 0.801·40-s + 1.50·41-s + 1.53·43-s + 0.232·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995865269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995865269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 1.16T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 + 9.29T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 0.751T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.689965113298439601886441344832, −7.923912239219640874943883581538, −7.17123270852008483250128031172, −6.02654755483831957265889069629, −5.68704047609438096562580052827, −4.57407946132494395948009944629, −4.01441064267072411827661078048, −3.44994660677369261711210148473, −2.24088906098804488836170591827, −0.75116451254716444647303943173,
0.75116451254716444647303943173, 2.24088906098804488836170591827, 3.44994660677369261711210148473, 4.01441064267072411827661078048, 4.57407946132494395948009944629, 5.68704047609438096562580052827, 6.02654755483831957265889069629, 7.17123270852008483250128031172, 7.923912239219640874943883581538, 8.689965113298439601886441344832
