| L(s) = 1 | − 1.64·2-s − 3-s + 0.700·4-s + 2.37·5-s + 1.64·6-s − 1.38·7-s + 2.13·8-s + 9-s − 3.90·10-s + 2.70·11-s − 0.700·12-s + 4.44·13-s + 2.27·14-s − 2.37·15-s − 4.91·16-s + 6.80·17-s − 1.64·18-s + 2.58·19-s + 1.66·20-s + 1.38·21-s − 4.44·22-s − 0.161·23-s − 2.13·24-s + 0.635·25-s − 7.31·26-s − 27-s − 0.971·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.350·4-s + 1.06·5-s + 0.670·6-s − 0.524·7-s + 0.754·8-s + 0.333·9-s − 1.23·10-s + 0.815·11-s − 0.202·12-s + 1.23·13-s + 0.609·14-s − 0.612·15-s − 1.22·16-s + 1.64·17-s − 0.387·18-s + 0.592·19-s + 0.371·20-s + 0.302·21-s − 0.948·22-s − 0.0337·23-s − 0.435·24-s + 0.127·25-s − 1.43·26-s − 0.192·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059432660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059432660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 677 | \( 1 - T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 6.80T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 0.161T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 6.87T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 + 0.961T + 73T^{2} \) |
| 79 | \( 1 + 0.369T + 79T^{2} \) |
| 83 | \( 1 - 0.545T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 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
−9.379661513554115812841040774956, −8.494084682161202794850811056485, −7.74607793414956258065842371343, −6.74615159261895055950433171021, −6.10856307718422621502571340701, −5.42576247737635105608055280134, −4.25411898089153904938113094946, −3.14927607221722378550305341288, −1.59806340870118909583869311632, −0.950269146986692268092185454903,
0.950269146986692268092185454903, 1.59806340870118909583869311632, 3.14927607221722378550305341288, 4.25411898089153904938113094946, 5.42576247737635105608055280134, 6.10856307718422621502571340701, 6.74615159261895055950433171021, 7.74607793414956258065842371343, 8.494084682161202794850811056485, 9.379661513554115812841040774956
