| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−9.168085073292481157063770000509, −8.581135683947151910351653901816, −7.52125503193221693145217635813, −6.41465845896198861882394229388, −5.91901022245231424153446867954, −4.80199037379022078874381258374, −3.51786857718362026378215359112, −2.41223286146436620098563852173, −1.98343232508317552865563173237, 0,
1.98343232508317552865563173237, 2.41223286146436620098563852173, 3.51786857718362026378215359112, 4.80199037379022078874381258374, 5.91901022245231424153446867954, 6.41465845896198861882394229388, 7.52125503193221693145217635813, 8.581135683947151910351653901816, 9.168085073292481157063770000509
