| L(s) = 1 | − 4.24·2-s + 9.99·4-s − 11·7-s − 8.48·8-s + 16.9·11-s − 29·13-s + 46.6·14-s − 44.0·16-s + 50.9·17-s + 29·19-s − 71.9·22-s − 84.8·23-s + 123.·26-s − 109.·28-s + 271.·29-s − 268·31-s + 254.·32-s − 215.·34-s − 83·37-s − 123.·38-s − 271.·41-s + 232·43-s + 169.·44-s + 360·46-s + 390.·47-s − 222·49-s − 289.·52-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s − 0.593·7-s − 0.374·8-s + 0.465·11-s − 0.618·13-s + 0.890·14-s − 0.687·16-s + 0.726·17-s + 0.350·19-s − 0.697·22-s − 0.769·23-s + 0.928·26-s − 0.742·28-s + 1.73·29-s − 1.55·31-s + 1.40·32-s − 1.08·34-s − 0.368·37-s − 0.525·38-s − 1.03·41-s + 0.822·43-s + 0.581·44-s + 1.15·46-s + 1.21·47-s − 0.647·49-s − 0.773·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.24T + 8T^{2} \) |
| 7 | \( 1 + 11T + 343T^{2} \) |
| 11 | \( 1 - 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 268T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 232T + 7.95e4T^{2} \) |
| 47 | \( 1 - 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 767T + 2.26e5T^{2} \) |
| 67 | \( 1 - 511T + 3.00e5T^{2} \) |
| 71 | \( 1 - 712.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 137T + 3.89e5T^{2} \) |
| 79 | \( 1 + 475T + 4.93e5T^{2} \) |
| 83 | \( 1 + 576.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 254.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 821T + 9.12e5T^{2} \) |
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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
−9.752970009954713032302948319623, −8.884288458983274553116788075008, −8.083844281435676251140585582200, −7.21997759836997549781354511058, −6.49657974974853502813010212677, −5.24975331191266595583226346327, −3.83921208981546362992786668310, −2.50407051501345297346275278927, −1.21665495723127506118951012085, 0,
1.21665495723127506118951012085, 2.50407051501345297346275278927, 3.83921208981546362992786668310, 5.24975331191266595583226346327, 6.49657974974853502813010212677, 7.21997759836997549781354511058, 8.083844281435676251140585582200, 8.884288458983274553116788075008, 9.752970009954713032302948319623
