L(s) = 1 | − 1.28·2-s − 3-s − 0.349·4-s − 0.580·5-s + 1.28·6-s − 1.93·7-s + 3.01·8-s + 9-s + 0.746·10-s + 1.58·11-s + 0.349·12-s − 0.419·13-s + 2.48·14-s + 0.580·15-s − 3.17·16-s + 17-s − 1.28·18-s − 2.91·19-s + 0.203·20-s + 1.93·21-s − 2.03·22-s + 8.95·23-s − 3.01·24-s − 4.66·25-s + 0.539·26-s − 27-s + 0.675·28-s + ⋯ |
L(s) = 1 | − 0.908·2-s − 0.577·3-s − 0.174·4-s − 0.259·5-s + 0.524·6-s − 0.730·7-s + 1.06·8-s + 0.333·9-s + 0.235·10-s + 0.477·11-s + 0.100·12-s − 0.116·13-s + 0.663·14-s + 0.149·15-s − 0.794·16-s + 0.242·17-s − 0.302·18-s − 0.669·19-s + 0.0454·20-s + 0.421·21-s − 0.433·22-s + 1.86·23-s − 0.616·24-s − 0.932·25-s + 0.105·26-s − 0.192·27-s + 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6710236084\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6710236084\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 5 | \( 1 + 0.580T + 5T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + 0.419T + 13T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 7.92T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 5.25T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 0.696T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 + 0.402T + 83T^{2} \) |
| 89 | \( 1 - 3.07T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.74513121629445769440568764412, −7.36762222602336481266285501066, −6.42808257499055701775619353648, −6.03654366017180508831130783328, −4.86762745137802303334562130238, −4.44598242302360763476220549521, −3.55172214553040665073804419669, −2.56966692603741067453918196970, −1.31137411632102639434189292657, −0.54536192950425027064850548824,
0.54536192950425027064850548824, 1.31137411632102639434189292657, 2.56966692603741067453918196970, 3.55172214553040665073804419669, 4.44598242302360763476220549521, 4.86762745137802303334562130238, 6.03654366017180508831130783328, 6.42808257499055701775619353648, 7.36762222602336481266285501066, 7.74513121629445769440568764412