| L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s − 11-s + 2·13-s − 15-s + 5·17-s − 19-s + 3·21-s + 2·23-s + 25-s − 5·27-s + 3·29-s + 31-s − 33-s − 3·35-s + 37-s + 2·39-s − 2·41-s − 4·43-s + 2·45-s + 10·47-s + 2·49-s + 5·51-s + 13·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s + 1.21·17-s − 0.229·19-s + 0.654·21-s + 0.417·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.179·31-s − 0.174·33-s − 0.507·35-s + 0.164·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.45·47-s + 2/7·49-s + 0.700·51-s + 1.78·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.404350557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.404350557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.498764209085891433405348719746, −7.906555960685250416270964382402, −7.43214912340271834366960248867, −6.29863729391942159021444102650, −5.45517279069279365928218794037, −4.78546138476134529636182608235, −3.79465808042411714576052742293, −3.05932231001616320126259234264, −2.08582402133298538904452506072, −0.917962735436832424144253192805,
0.917962735436832424144253192805, 2.08582402133298538904452506072, 3.05932231001616320126259234264, 3.79465808042411714576052742293, 4.78546138476134529636182608235, 5.45517279069279365928218794037, 6.29863729391942159021444102650, 7.43214912340271834366960248867, 7.906555960685250416270964382402, 8.498764209085891433405348719746
