L(s) = 1 | − 3-s + 1.53·5-s + 2.90·7-s + 9-s − 11-s − 13-s − 1.53·15-s + 1.37·17-s − 6.41·19-s − 2.90·21-s − 2.90·23-s − 2.65·25-s − 27-s + 4.88·29-s + 1.78·31-s + 33-s + 4.44·35-s + 4.75·37-s + 39-s − 10.6·41-s + 0.130·43-s + 1.53·45-s − 8·47-s + 1.44·49-s − 1.37·51-s + 6·53-s − 1.53·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.684·5-s + 1.09·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.395·15-s + 0.333·17-s − 1.47·19-s − 0.634·21-s − 0.606·23-s − 0.531·25-s − 0.192·27-s + 0.906·29-s + 0.321·31-s + 0.174·33-s + 0.751·35-s + 0.781·37-s + 0.160·39-s − 1.66·41-s + 0.0199·43-s + 0.228·45-s − 1.16·47-s + 0.206·49-s − 0.192·51-s + 0.824·53-s − 0.206·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 0.130T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.31T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 - 4.75T + 97T^{2} \) |
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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
−7.68014454211963833367400770416, −6.80455139322290678856602803630, −6.09141764582217689460438787676, −5.58805565548896183386292568115, −4.66212273762110088978351166369, −4.38040053563062642991547789631, −3.06152484628195239368786027641, −2.04289724047551387210865041518, −1.45641723301485362132118641730, 0,
1.45641723301485362132118641730, 2.04289724047551387210865041518, 3.06152484628195239368786027641, 4.38040053563062642991547789631, 4.66212273762110088978351166369, 5.58805565548896183386292568115, 6.09141764582217689460438787676, 6.80455139322290678856602803630, 7.68014454211963833367400770416