L(s) = 1 | − 2.82·3-s − 0.343·5-s − 4.77·7-s + 4.98·9-s + 11-s + 1.86·13-s + 0.970·15-s − 5.69·17-s + 19-s + 13.4·21-s + 1.06·23-s − 4.88·25-s − 5.60·27-s − 9.31·29-s − 2.97·31-s − 2.82·33-s + 1.64·35-s − 3.54·37-s − 5.26·39-s + 2.82·41-s − 8.09·43-s − 1.71·45-s − 3.00·47-s + 15.7·49-s + 16.0·51-s − 4.33·53-s − 0.343·55-s + ⋯ |
L(s) = 1 | − 1.63·3-s − 0.153·5-s − 1.80·7-s + 1.66·9-s + 0.301·11-s + 0.516·13-s + 0.250·15-s − 1.38·17-s + 0.229·19-s + 2.94·21-s + 0.221·23-s − 0.976·25-s − 1.07·27-s − 1.72·29-s − 0.533·31-s − 0.491·33-s + 0.277·35-s − 0.583·37-s − 0.843·39-s + 0.440·41-s − 1.23·43-s − 0.255·45-s − 0.438·47-s + 2.25·49-s + 2.25·51-s − 0.595·53-s − 0.0463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2381668744\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2381668744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.343T + 5T^{2} \) |
| 7 | \( 1 + 4.77T + 7T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 3.54T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 8.09T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 4.33T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 3.40T + 83T^{2} \) |
| 89 | \( 1 - 0.0562T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.895433888206134493804804515983, −7.54009232262622717309616066580, −6.86102808152844386689559173610, −6.24257885975875102244819084908, −5.88805238093388349273076849601, −4.92286150330770924911221963557, −3.98090417977121357932676660361, −3.28638151726355241030223251800, −1.79915736880155457527395826386, −0.30521649179204643516154821287,
0.30521649179204643516154821287, 1.79915736880155457527395826386, 3.28638151726355241030223251800, 3.98090417977121357932676660361, 4.92286150330770924911221963557, 5.88805238093388349273076849601, 6.24257885975875102244819084908, 6.86102808152844386689559173610, 7.54009232262622717309616066580, 8.895433888206134493804804515983