| L(s) = 1 | + 3·3-s − 21.1·5-s + 7·7-s + 9·9-s + 11.1·11-s − 26·13-s − 63.3·15-s + 67.3·17-s − 42.2·19-s + 21·21-s + 122.·23-s + 320.·25-s + 27·27-s + 16.2·29-s − 279.·31-s + 33.3·33-s − 147.·35-s + 123.·37-s − 78·39-s − 32.2·41-s + 281.·43-s − 190.·45-s − 250.·47-s + 49·49-s + 202.·51-s + 27.8·53-s − 234.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·5-s + 0.377·7-s + 0.333·9-s + 0.304·11-s − 0.554·13-s − 1.09·15-s + 0.960·17-s − 0.509·19-s + 0.218·21-s + 1.10·23-s + 2.56·25-s + 0.192·27-s + 0.103·29-s − 1.61·31-s + 0.175·33-s − 0.713·35-s + 0.550·37-s − 0.320·39-s − 0.122·41-s + 0.999·43-s − 0.629·45-s − 0.778·47-s + 0.142·49-s + 0.554·51-s + 0.0721·53-s − 0.575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 21.1T + 125T^{2} \) |
| 11 | \( 1 - 11.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 16.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 123.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 32.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 27.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 677.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 73.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 845.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 739.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 250.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 699.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 489.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−8.765773474549285000849657438062, −7.78807984265371307780839646689, −7.59524243194691572372439091742, −6.67220135564744767305174794909, −5.21589298257654618303128450704, −4.36590639110758336081693397308, −3.63382591770120307057798874907, −2.80499980690278310610287067225, −1.26080494945028391218788277094, 0,
1.26080494945028391218788277094, 2.80499980690278310610287067225, 3.63382591770120307057798874907, 4.36590639110758336081693397308, 5.21589298257654618303128450704, 6.67220135564744767305174794909, 7.59524243194691572372439091742, 7.78807984265371307780839646689, 8.765773474549285000849657438062
