| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 11-s − 5·13-s + 2·17-s − 4·21-s − 8·23-s − 5·25-s + 4·27-s + 5·29-s − 2·33-s − 2·37-s + 10·39-s + 2·41-s − 11·43-s + 9·47-s − 3·49-s − 4·51-s + 2·53-s + 14·59-s + 7·61-s + 2·63-s − 2·67-s + 16·69-s − 3·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.485·17-s − 0.872·21-s − 1.66·23-s − 25-s + 0.769·27-s + 0.928·29-s − 0.348·33-s − 0.328·37-s + 1.60·39-s + 0.312·41-s − 1.67·43-s + 1.31·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 1.82·59-s + 0.896·61-s + 0.251·63-s − 0.244·67-s + 1.92·69-s − 0.356·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.44328743795491, −14.66780246370801, −14.28467532004677, −13.95899174043508, −13.09137375968877, −12.47122866599706, −11.93930235042538, −11.67767315089525, −11.36143783163376, −10.39120281610249, −10.06821835599959, −9.760502290346871, −8.614962127304729, −8.389144990489165, −7.452744084843014, −7.244363498978471, −6.277068346021759, −5.945262292861671, −5.236741388238639, −4.821703110235150, −4.210242662804741, −3.435039894087968, −2.406233885089577, −1.847287665696157, −0.8362709909396113, 0,
0.8362709909396113, 1.847287665696157, 2.406233885089577, 3.435039894087968, 4.210242662804741, 4.821703110235150, 5.236741388238639, 5.945262292861671, 6.277068346021759, 7.244363498978471, 7.452744084843014, 8.389144990489165, 8.614962127304729, 9.760502290346871, 10.06821835599959, 10.39120281610249, 11.36143783163376, 11.67767315089525, 11.93930235042538, 12.47122866599706, 13.09137375968877, 13.95899174043508, 14.28467532004677, 14.66780246370801, 15.44328743795491
