L(s) = 1 | − 3.24·2-s + 5.49·3-s + 2.53·4-s + 16.0·5-s − 17.8·6-s + 17.7·8-s + 3.16·9-s − 52.2·10-s − 11·11-s + 13.9·12-s − 35.3·13-s + 88.4·15-s − 77.8·16-s − 40.4·17-s − 10.2·18-s − 118.·19-s + 40.7·20-s + 35.7·22-s − 174.·23-s + 97.4·24-s + 134.·25-s + 114.·26-s − 130.·27-s − 262.·29-s − 286.·30-s + 36.1·31-s + 110.·32-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 1.05·3-s + 0.316·4-s + 1.43·5-s − 1.21·6-s + 0.784·8-s + 0.117·9-s − 1.65·10-s − 0.301·11-s + 0.334·12-s − 0.754·13-s + 1.52·15-s − 1.21·16-s − 0.577·17-s − 0.134·18-s − 1.42·19-s + 0.455·20-s + 0.345·22-s − 1.58·23-s + 0.828·24-s + 1.07·25-s + 0.865·26-s − 0.933·27-s − 1.68·29-s − 1.74·30-s + 0.209·31-s + 0.611·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 3.24T + 8T^{2} \) |
| 3 | \( 1 - 5.49T + 27T^{2} \) |
| 5 | \( 1 - 16.0T + 125T^{2} \) |
| 13 | \( 1 + 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 36.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 35.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 979.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 750.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 395.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 582.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 957.T + 9.12e5T^{2} \) |
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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
−9.794605231442021170903441657025, −9.117147517598968188007689853891, −8.433856634033415129458712144440, −7.66141695559602901058534659429, −6.52050037129876300025702671199, −5.40437862728335527339831346095, −4.05487967977970461865797613183, −2.29851406344578814534564306992, −1.95910253702569987459652346742, 0,
1.95910253702569987459652346742, 2.29851406344578814534564306992, 4.05487967977970461865797613183, 5.40437862728335527339831346095, 6.52050037129876300025702671199, 7.66141695559602901058534659429, 8.433856634033415129458712144440, 9.117147517598968188007689853891, 9.794605231442021170903441657025