| L(s) = 1 | + 5·5-s + 34·7-s + 16·11-s + 58·13-s + 70·17-s − 4·19-s − 134·23-s + 25·25-s + 242·29-s − 100·31-s + 170·35-s − 438·37-s + 138·41-s − 178·43-s + 22·47-s + 813·49-s − 162·53-s + 80·55-s − 268·59-s + 250·61-s + 290·65-s − 422·67-s − 852·71-s + 306·73-s + 544·77-s + 456·79-s + 434·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.83·7-s + 0.438·11-s + 1.23·13-s + 0.998·17-s − 0.0482·19-s − 1.21·23-s + 1/5·25-s + 1.54·29-s − 0.579·31-s + 0.821·35-s − 1.94·37-s + 0.525·41-s − 0.631·43-s + 0.0682·47-s + 2.37·49-s − 0.419·53-s + 0.196·55-s − 0.591·59-s + 0.524·61-s + 0.553·65-s − 0.769·67-s − 1.42·71-s + 0.490·73-s + 0.805·77-s + 0.649·79-s + 0.573·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.171541832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.171541832\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 438 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 22 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 422 T + p^{3} T^{2} \) |
| 71 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 - 306 T + p^{3} T^{2} \) |
| 79 | \( 1 - 456 T + p^{3} T^{2} \) |
| 83 | \( 1 - 434 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1378 T + p^{3} T^{2} \) |
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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
−10.22572480046913745011930392958, −8.947607958223266007249640291711, −8.335377338324811771435674064641, −7.59054497170985846597906126568, −6.36464635179123227994521505831, −5.49226044370723787902006471980, −4.60805128938907848460513995844, −3.52565889202379224030494240673, −1.92193744145234397781099933324, −1.16728557410045569590268749643,
1.16728557410045569590268749643, 1.92193744145234397781099933324, 3.52565889202379224030494240673, 4.60805128938907848460513995844, 5.49226044370723787902006471980, 6.36464635179123227994521505831, 7.59054497170985846597906126568, 8.335377338324811771435674064641, 8.947607958223266007249640291711, 10.22572480046913745011930392958
