| L(s) = 1 | − 3·3-s + 5·5-s − 20·7-s + 9·9-s − 24·11-s − 74·13-s − 15·15-s + 54·17-s − 124·19-s + 60·21-s + 120·23-s + 25·25-s − 27·27-s + 78·29-s − 200·31-s + 72·33-s − 100·35-s + 70·37-s + 222·39-s + 330·41-s + 92·43-s + 45·45-s + 24·47-s + 57·49-s − 162·51-s − 450·53-s − 120·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.07·7-s + 1/3·9-s − 0.657·11-s − 1.57·13-s − 0.258·15-s + 0.770·17-s − 1.49·19-s + 0.623·21-s + 1.08·23-s + 1/5·25-s − 0.192·27-s + 0.499·29-s − 1.15·31-s + 0.379·33-s − 0.482·35-s + 0.311·37-s + 0.911·39-s + 1.25·41-s + 0.326·43-s + 0.149·45-s + 0.0744·47-s + 0.166·49-s − 0.444·51-s − 1.16·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9329722687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9329722687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.733629609714883238692302085640, −9.072101551103016518387292291749, −7.78895620021114900221949216110, −6.99856254987306546810255407075, −6.20169051729730916277908063786, −5.35438382083019280139378531551, −4.50526366285400123004103083523, −3.13613700503052053857312340941, −2.19039060817134113378631151261, −0.50764496768865082402193622764,
0.50764496768865082402193622764, 2.19039060817134113378631151261, 3.13613700503052053857312340941, 4.50526366285400123004103083523, 5.35438382083019280139378531551, 6.20169051729730916277908063786, 6.99856254987306546810255407075, 7.78895620021114900221949216110, 9.072101551103016518387292291749, 9.733629609714883238692302085640
