| L(s) = 1 | − 4.18·3-s − 10.8·5-s − 7·7-s − 9.50·9-s + 11·11-s + 43.5·13-s + 45.3·15-s − 10.4·17-s − 92.6·19-s + 29.2·21-s + 102.·23-s − 7.17·25-s + 152.·27-s + 49.3·29-s + 256.·31-s − 46.0·33-s + 75.9·35-s + 88.0·37-s − 182.·39-s − 409.·41-s − 12.4·43-s + 103.·45-s + 638.·47-s + 49·49-s + 43.5·51-s + 51.5·53-s − 119.·55-s + ⋯ |
| L(s) = 1 | − 0.804·3-s − 0.970·5-s − 0.377·7-s − 0.352·9-s + 0.301·11-s + 0.928·13-s + 0.781·15-s − 0.148·17-s − 1.11·19-s + 0.304·21-s + 0.925·23-s − 0.0573·25-s + 1.08·27-s + 0.315·29-s + 1.48·31-s − 0.242·33-s + 0.366·35-s + 0.391·37-s − 0.747·39-s − 1.55·41-s − 0.0441·43-s + 0.341·45-s + 1.98·47-s + 0.142·49-s + 0.119·51-s + 0.133·53-s − 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 4.18T + 27T^{2} \) |
| 5 | \( 1 + 10.8T + 125T^{2} \) |
| 13 | \( 1 - 43.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 256.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 88.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 12.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 638.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 51.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 355.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 8.37T + 3.57e5T^{2} \) |
| 73 | \( 1 - 339.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 989.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 303.T + 9.12e5T^{2} \) |
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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
−8.618075045602317357750371021483, −8.381140976590651581295097057732, −7.07857992633878449315745888561, −6.44286233693120985926445782931, −5.65827006878352880285790610542, −4.56598705897852371945552316594, −3.79238949587037540849005951893, −2.71573902037702056966804514557, −1.01824625340668746304442824985, 0,
1.01824625340668746304442824985, 2.71573902037702056966804514557, 3.79238949587037540849005951893, 4.56598705897852371945552316594, 5.65827006878352880285790610542, 6.44286233693120985926445782931, 7.07857992633878449315745888561, 8.381140976590651581295097057732, 8.618075045602317357750371021483
