| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 11-s + 2·13-s + 15-s + 17-s + 4·19-s + 4·21-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s + 4·35-s + 6·37-s − 2·39-s + 10·41-s + 4·43-s − 45-s + 9·49-s − 51-s + 14·53-s − 55-s − 4·57-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.676·35-s + 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s + 1.92·53-s − 0.134·55-s − 0.529·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.33646958967953, −12.86658262967737, −12.48900227604813, −11.97384337312712, −11.57040235458403, −11.05711536222740, −10.67036970187890, −9.986862635545063, −9.601249733831141, −9.326648102511916, −8.698513012172364, −8.042273663019665, −7.573058764130975, −6.964319128177161, −6.688114443394460, −6.062480113664676, −5.624783611320322, −5.260270233861783, −4.285565339748086, −3.910781830427228, −3.553958217749829, −2.759230006698743, −2.371037523584743, −1.153591037014001, −0.8325089889550954, 0,
0.8325089889550954, 1.153591037014001, 2.371037523584743, 2.759230006698743, 3.553958217749829, 3.910781830427228, 4.285565339748086, 5.260270233861783, 5.624783611320322, 6.062480113664676, 6.688114443394460, 6.964319128177161, 7.573058764130975, 8.042273663019665, 8.698513012172364, 9.326648102511916, 9.601249733831141, 9.986862635545063, 10.67036970187890, 11.05711536222740, 11.57040235458403, 11.97384337312712, 12.48900227604813, 12.86658262967737, 13.33646958967953
