| L(s) = 1 | − 2.06·2-s − 1.75·3-s + 2.27·4-s − 3.28·5-s + 3.63·6-s + 1.71·7-s − 0.561·8-s + 0.0971·9-s + 6.79·10-s − 11-s − 3.99·12-s − 0.888·13-s − 3.53·14-s + 5.78·15-s − 3.38·16-s − 3.91·17-s − 0.200·18-s + 4.66·19-s − 7.47·20-s − 3.01·21-s + 2.06·22-s − 1.83·23-s + 0.988·24-s + 5.82·25-s + 1.83·26-s + 5.10·27-s + 3.88·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.01·3-s + 1.13·4-s − 1.47·5-s + 1.48·6-s + 0.646·7-s − 0.198·8-s + 0.0323·9-s + 2.14·10-s − 0.301·11-s − 1.15·12-s − 0.246·13-s − 0.944·14-s + 1.49·15-s − 0.845·16-s − 0.949·17-s − 0.0473·18-s + 1.07·19-s − 1.67·20-s − 0.656·21-s + 0.440·22-s − 0.383·23-s + 0.201·24-s + 1.16·25-s + 0.360·26-s + 0.983·27-s + 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 - 3.10T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 0.818T + 67T^{2} \) |
| 71 | \( 1 - 4.11T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 + 6.15T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.047113148169293081922099943198, −8.213878743089975248992188537603, −7.70853068141519365972475909515, −7.07927426387947725247270225488, −6.01893305381193129233355604658, −4.87132852715583443293299451157, −4.18232721808901713890724682254, −2.61764246364544445667190074327, −1.00558589983449999817810092692, 0,
1.00558589983449999817810092692, 2.61764246364544445667190074327, 4.18232721808901713890724682254, 4.87132852715583443293299451157, 6.01893305381193129233355604658, 7.07927426387947725247270225488, 7.70853068141519365972475909515, 8.213878743089975248992188537603, 9.047113148169293081922099943198
