L(s) = 1 | − 2.53·5-s + 7-s − 4.91·11-s − 0.165·13-s − 6.84·17-s + 19-s − 2.16·23-s + 1.43·25-s − 4.70·29-s − 7.07·31-s − 2.53·35-s − 0.746·37-s − 3.07·41-s − 3.88·47-s + 49-s + 0.702·53-s + 12.4·55-s + 13.3·59-s + 6·61-s + 0.419·65-s − 13.0·67-s + 8.26·71-s − 5.05·73-s − 4.91·77-s + 9.22·79-s − 8.19·83-s + 17.3·85-s + ⋯ |
L(s) = 1 | − 1.13·5-s + 0.377·7-s − 1.48·11-s − 0.0459·13-s − 1.66·17-s + 0.229·19-s − 0.451·23-s + 0.287·25-s − 0.873·29-s − 1.27·31-s − 0.428·35-s − 0.122·37-s − 0.480·41-s − 0.566·47-s + 0.142·49-s + 0.0964·53-s + 1.68·55-s + 1.73·59-s + 0.768·61-s + 0.0520·65-s − 1.59·67-s + 0.981·71-s − 0.591·73-s − 0.559·77-s + 1.03·79-s − 0.899·83-s + 1.88·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4880124089\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4880124089\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 + 0.165T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.746T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 - 0.702T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.26T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.56574373467446865642442213231, −7.30529304155128337310313550832, −6.40945460484058457153849874702, −5.45272280538112849703193518582, −4.95106684766348840680704185229, −4.14236618183931777645538345078, −3.57501215913349413776621631070, −2.56399640259234639467548118934, −1.86314391387657025371408245253, −0.31422576490895026381145820510,
0.31422576490895026381145820510, 1.86314391387657025371408245253, 2.56399640259234639467548118934, 3.57501215913349413776621631070, 4.14236618183931777645538345078, 4.95106684766348840680704185229, 5.45272280538112849703193518582, 6.40945460484058457153849874702, 7.30529304155128337310313550832, 7.56574373467446865642442213231