| L(s) = 1 | − 4.62·3-s + 17.8·5-s + 13.8·7-s − 5.59·9-s + 2.18·11-s − 46.3·13-s − 82.7·15-s − 18.6·17-s + 122.·19-s − 64.1·21-s + 134.·23-s + 194.·25-s + 150.·27-s − 84.5·29-s − 31.5·31-s − 10.1·33-s + 248.·35-s − 126.·37-s + 214.·39-s + 210.·41-s + 168.·43-s − 100.·45-s + 182.·47-s − 150.·49-s + 86.2·51-s − 37.0·53-s + 39.1·55-s + ⋯ |
| L(s) = 1 | − 0.890·3-s + 1.59·5-s + 0.749·7-s − 0.207·9-s + 0.0599·11-s − 0.989·13-s − 1.42·15-s − 0.266·17-s + 1.47·19-s − 0.667·21-s + 1.21·23-s + 1.55·25-s + 1.07·27-s − 0.541·29-s − 0.182·31-s − 0.0533·33-s + 1.19·35-s − 0.560·37-s + 0.880·39-s + 0.801·41-s + 0.598·43-s − 0.331·45-s + 0.567·47-s − 0.438·49-s + 0.236·51-s − 0.0958·53-s + 0.0959·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.207526728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.207526728\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4.62T + 27T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 11 | \( 1 - 2.18T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 37.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 624.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 246.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 943.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−9.481495775563623732639962675612, −9.100174846290486424359952704520, −7.76105912531889237885744736693, −6.87620716681115594132372500429, −5.95473055073803168537629884883, −5.27133530803626868856683901837, −4.81487967790560207153667663445, −2.99304256882878182906510455231, −1.95025969989089373048976509320, −0.848821938234964779563975377470,
0.848821938234964779563975377470, 1.95025969989089373048976509320, 2.99304256882878182906510455231, 4.81487967790560207153667663445, 5.27133530803626868856683901837, 5.95473055073803168537629884883, 6.87620716681115594132372500429, 7.76105912531889237885744736693, 9.100174846290486424359952704520, 9.481495775563623732639962675612
