L(s) = 1 | + 5-s + 5·7-s + 4·13-s + 3·17-s + 4·19-s − 23-s + 25-s − 29-s − 31-s + 5·35-s − 37-s − 11·41-s − 4·43-s + 6·47-s + 18·49-s − 53-s − 59-s + 6·61-s + 4·65-s + 9·67-s + 13·71-s − 16·79-s + 9·83-s + 3·85-s − 8·89-s + 20·91-s + 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.88·7-s + 1.10·13-s + 0.727·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.185·29-s − 0.179·31-s + 0.845·35-s − 0.164·37-s − 1.71·41-s − 0.609·43-s + 0.875·47-s + 18/7·49-s − 0.137·53-s − 0.130·59-s + 0.768·61-s + 0.496·65-s + 1.09·67-s + 1.54·71-s − 1.80·79-s + 0.987·83-s + 0.325·85-s − 0.847·89-s + 2.09·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.919896633\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.919896633\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.82907108525449, −15.35357020497517, −14.70057473866886, −14.23641096277443, −13.79061159502061, −13.37493548454666, −12.50447780965310, −11.89611278071138, −11.39346233686850, −11.00452811643123, −10.30006348624295, −9.800208355116328, −8.932540890136984, −8.443104073852636, −7.988003760086389, −7.352450036402266, −6.658320094739152, −5.750342971558451, −5.333892197084605, −4.806433530168079, −3.928046259041324, −3.298683641499180, −2.205151663781755, −1.550795214607528, −0.9457395770098853,
0.9457395770098853, 1.550795214607528, 2.205151663781755, 3.298683641499180, 3.928046259041324, 4.806433530168079, 5.333892197084605, 5.750342971558451, 6.658320094739152, 7.352450036402266, 7.988003760086389, 8.443104073852636, 8.932540890136984, 9.800208355116328, 10.30006348624295, 11.00452811643123, 11.39346233686850, 11.89611278071138, 12.50447780965310, 13.37493548454666, 13.79061159502061, 14.23641096277443, 14.70057473866886, 15.35357020497517, 15.82907108525449