L(s) = 1 | + 5-s − 7-s − 4·11-s − 13-s + 6·17-s + 6·23-s + 25-s − 6·29-s + 6·31-s − 35-s − 6·37-s − 2·41-s + 49-s − 4·55-s + 4·59-s + 2·61-s − 65-s + 2·67-s − 8·71-s − 8·73-s + 4·77-s − 12·83-s + 6·85-s − 14·89-s + 91-s − 8·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.07·31-s − 0.169·35-s − 0.986·37-s − 0.312·41-s + 1/7·49-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.124·65-s + 0.244·67-s − 0.949·71-s − 0.936·73-s + 0.455·77-s − 1.31·83-s + 0.650·85-s − 1.48·89-s + 0.104·91-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.13596948177586, −14.93905075052566, −14.11539658060727, −13.76767968464586, −13.02473732252729, −12.84371738458792, −12.20988107443035, −11.59589068524968, −10.98580838552661, −10.32264423302707, −10.02914474119070, −9.527668206503909, −8.780474644422962, −8.303481251147292, −7.549884425771600, −7.198759910198642, −6.511960200642165, −5.653018281604385, −5.423632631252652, −4.799985850496793, −3.941465842032746, −3.049863673258959, −2.830653010130086, −1.848957023126813, −1.025794383045675, 0,
1.025794383045675, 1.848957023126813, 2.830653010130086, 3.049863673258959, 3.941465842032746, 4.799985850496793, 5.423632631252652, 5.653018281604385, 6.511960200642165, 7.198759910198642, 7.549884425771600, 8.303481251147292, 8.780474644422962, 9.527668206503909, 10.02914474119070, 10.32264423302707, 10.98580838552661, 11.59589068524968, 12.20988107443035, 12.84371738458792, 13.02473732252729, 13.76767968464586, 14.11539658060727, 14.93905075052566, 15.13596948177586