| L(s) = 1 | + 2-s + 4-s − 0.618·5-s − 7-s + 8-s − 0.618·10-s − 1.61·13-s − 14-s + 16-s − 6.23·17-s + 5.85·19-s − 0.618·20-s + 5.09·23-s − 4.61·25-s − 1.61·26-s − 28-s + 4.09·29-s − 2.14·31-s + 32-s − 6.23·34-s + 0.618·35-s − 9·37-s + 5.85·38-s − 0.618·40-s − 2.61·41-s + 4.85·43-s + 5.09·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.276·5-s − 0.377·7-s + 0.353·8-s − 0.195·10-s − 0.448·13-s − 0.267·14-s + 0.250·16-s − 1.51·17-s + 1.34·19-s − 0.138·20-s + 1.06·23-s − 0.923·25-s − 0.317·26-s − 0.188·28-s + 0.759·29-s − 0.385·31-s + 0.176·32-s − 1.06·34-s + 0.104·35-s − 1.47·37-s + 0.949·38-s − 0.0977·40-s − 0.408·41-s + 0.740·43-s + 0.750·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 3.61T + 47T^{2} \) |
| 53 | \( 1 + 0.381T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 + 8.32T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−7.43357179324622289348297547548, −6.90590655331837434171981229148, −6.29684320049962768226148620850, −5.34587748907023781384410654009, −4.84331762444433056252434022589, −4.00681382970800957395810224352, −3.24018071982111091267034539587, −2.53123689111555982433597224433, −1.46227084712496427512624888223, 0,
1.46227084712496427512624888223, 2.53123689111555982433597224433, 3.24018071982111091267034539587, 4.00681382970800957395810224352, 4.84331762444433056252434022589, 5.34587748907023781384410654009, 6.29684320049962768226148620850, 6.90590655331837434171981229148, 7.43357179324622289348297547548
