L(s) = 1 | + 2.09·3-s − 3.36·5-s + 4.47·7-s + 1.39·9-s + 0.608·11-s − 1.03·13-s − 7.05·15-s − 3.06·17-s + 19-s + 9.36·21-s − 8.50·23-s + 6.34·25-s − 3.37·27-s − 7.27·29-s − 4.02·31-s + 1.27·33-s − 15.0·35-s − 4.31·37-s − 2.17·39-s + 4.15·41-s − 6.27·43-s − 4.68·45-s − 4.73·47-s + 12.9·49-s − 6.42·51-s − 6.98·53-s − 2.05·55-s + ⋯ |
L(s) = 1 | + 1.20·3-s − 1.50·5-s + 1.68·7-s + 0.463·9-s + 0.183·11-s − 0.288·13-s − 1.82·15-s − 0.743·17-s + 0.229·19-s + 2.04·21-s − 1.77·23-s + 1.26·25-s − 0.648·27-s − 1.35·29-s − 0.722·31-s + 0.222·33-s − 2.54·35-s − 0.708·37-s − 0.348·39-s + 0.649·41-s − 0.957·43-s − 0.698·45-s − 0.690·47-s + 1.85·49-s − 0.898·51-s − 0.958·53-s − 0.276·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 0.608T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 23 | \( 1 + 8.50T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 2.64T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 0.913T + 79T^{2} \) |
| 83 | \( 1 + 0.887T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.053295390819736421264066759032, −7.59915412272609210754211605222, −6.81045474533612778362640356898, −5.51436857343667272054156174522, −4.71776682411426978594457305887, −3.94509772289486138986031681741, −3.57271331904454812383877966050, −2.31787411629615358996354069388, −1.67354678314274588866472529600, 0,
1.67354678314274588866472529600, 2.31787411629615358996354069388, 3.57271331904454812383877966050, 3.94509772289486138986031681741, 4.71776682411426978594457305887, 5.51436857343667272054156174522, 6.81045474533612778362640356898, 7.59915412272609210754211605222, 8.053295390819736421264066759032