L(s) = 1 | − 2-s − 0.370·3-s + 4-s + 2.98·5-s + 0.370·6-s − 2.56·7-s − 8-s − 2.86·9-s − 2.98·10-s − 5.77·11-s − 0.370·12-s − 0.953·13-s + 2.56·14-s − 1.10·15-s + 16-s + 2.51·17-s + 2.86·18-s − 7.19·19-s + 2.98·20-s + 0.950·21-s + 5.77·22-s − 5.17·23-s + 0.370·24-s + 3.88·25-s + 0.953·26-s + 2.17·27-s − 2.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.213·3-s + 0.5·4-s + 1.33·5-s + 0.151·6-s − 0.969·7-s − 0.353·8-s − 0.954·9-s − 0.942·10-s − 1.74·11-s − 0.106·12-s − 0.264·13-s + 0.685·14-s − 0.285·15-s + 0.250·16-s + 0.610·17-s + 0.674·18-s − 1.65·19-s + 0.666·20-s + 0.207·21-s + 1.23·22-s − 1.07·23-s + 0.0756·24-s + 0.776·25-s + 0.187·26-s + 0.418·27-s − 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 269 | \( 1 + T \) |
good | 3 | \( 1 + 0.370T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 + 0.953T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 8.16T + 31T^{2} \) |
| 37 | \( 1 + 9.43T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + 9.50T + 47T^{2} \) |
| 53 | \( 1 + 7.51T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 + 5.86T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.650T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 8.27T + 83T^{2} \) |
| 89 | \( 1 - 2.89T + 89T^{2} \) |
| 97 | \( 1 - 0.956T + 97T^{2} \) |
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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
−10.22053309896049561464534792498, −9.753775148186898561456895727304, −8.632210486266637574638671021135, −7.918387694291356414119042394750, −6.45881248104667637849398023658, −6.02725912501540600442617313467, −5.02817614322771125511308549365, −2.99079563057587564427639663242, −2.20527310807920458245245374769, 0,
2.20527310807920458245245374769, 2.99079563057587564427639663242, 5.02817614322771125511308549365, 6.02725912501540600442617313467, 6.45881248104667637849398023658, 7.918387694291356414119042394750, 8.632210486266637574638671021135, 9.753775148186898561456895727304, 10.22053309896049561464534792498