| L(s) = 1 | + 0.869·3-s + 5.39·5-s + 27.1·7-s − 26.2·9-s − 19.5·11-s − 5.93·13-s + 4.68·15-s + 10.9·17-s − 15.0·19-s + 23.6·21-s + 23·23-s − 95.9·25-s − 46.3·27-s − 129.·29-s − 98.4·31-s − 16.9·33-s + 146.·35-s − 303.·37-s − 5.16·39-s + 32.8·41-s + 201.·43-s − 141.·45-s − 226.·47-s + 396.·49-s + 9.54·51-s − 75.1·53-s − 105.·55-s + ⋯ |
| L(s) = 1 | + 0.167·3-s + 0.482·5-s + 1.46·7-s − 0.971·9-s − 0.534·11-s − 0.126·13-s + 0.0807·15-s + 0.156·17-s − 0.181·19-s + 0.245·21-s + 0.208·23-s − 0.767·25-s − 0.330·27-s − 0.830·29-s − 0.570·31-s − 0.0895·33-s + 0.707·35-s − 1.34·37-s − 0.0211·39-s + 0.125·41-s + 0.716·43-s − 0.468·45-s − 0.703·47-s + 1.15·49-s + 0.0262·51-s − 0.194·53-s − 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 0.869T + 27T^{2} \) |
| 5 | \( 1 - 5.39T + 125T^{2} \) |
| 7 | \( 1 - 27.1T + 343T^{2} \) |
| 11 | \( 1 + 19.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.93T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 98.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 32.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 75.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 533.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 254.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 55.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 59.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 678.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−8.699791289671284302635538580939, −7.954130287233042989371379774823, −7.37023394859753013639162157551, −6.05852686529879409119463490593, −5.39989182351613352474731601110, −4.69699274959453350568210516867, −3.47672622399664893278054428752, −2.33344957230774340014025764665, −1.56892563849337081818485043626, 0,
1.56892563849337081818485043626, 2.33344957230774340014025764665, 3.47672622399664893278054428752, 4.69699274959453350568210516867, 5.39989182351613352474731601110, 6.05852686529879409119463490593, 7.37023394859753013639162157551, 7.954130287233042989371379774823, 8.699791289671284302635538580939
