L(s) = 1 | + 5-s + 7-s + 4·13-s − 3·17-s + 8·19-s − 4·23-s − 4·25-s + 8·29-s − 10·31-s + 35-s − 2·37-s + 6·41-s + 43-s − 13·47-s + 49-s − 8·53-s + 3·59-s + 4·65-s + 7·67-s + 6·71-s + 6·73-s − 8·79-s + 83-s − 3·85-s + 3·89-s + 4·91-s + 8·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.10·13-s − 0.727·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s + 1.48·29-s − 1.79·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s + 0.152·43-s − 1.89·47-s + 1/7·49-s − 1.09·53-s + 0.390·59-s + 0.496·65-s + 0.855·67-s + 0.712·71-s + 0.702·73-s − 0.900·79-s + 0.109·83-s − 0.325·85-s + 0.317·89-s + 0.419·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.886580508\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.886580508\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.14818057702236, −13.92140290720741, −13.33254657901481, −12.87442705541369, −12.26945022845575, −11.67436359124668, −11.18451634223508, −10.93760396579999, −10.05486899305893, −9.761934374869297, −9.173352699447449, −8.649079172307132, −8.026326753773170, −7.656706442083605, −6.910780444710249, −6.361654027562745, −5.841278184063201, −5.286835056448586, −4.764041003006345, −3.950170278270117, −3.491327275633787, −2.780208329362726, −1.913809498738170, −1.470934794746534, −0.5850999093292416,
0.5850999093292416, 1.470934794746534, 1.913809498738170, 2.780208329362726, 3.491327275633787, 3.950170278270117, 4.764041003006345, 5.286835056448586, 5.841278184063201, 6.361654027562745, 6.910780444710249, 7.656706442083605, 8.026326753773170, 8.649079172307132, 9.173352699447449, 9.761934374869297, 10.05486899305893, 10.93760396579999, 11.18451634223508, 11.67436359124668, 12.26945022845575, 12.87442705541369, 13.33254657901481, 13.92140290720741, 14.14818057702236