| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s + 2·11-s − 2·15-s − 2·17-s − 4·21-s + 6·23-s − 25-s + 27-s + 6·29-s − 4·31-s + 2·33-s + 8·35-s + 4·37-s − 2·41-s − 4·43-s − 2·45-s + 6·47-s + 9·49-s − 2·51-s − 2·53-s − 4·55-s + 12·59-s + 10·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.485·17-s − 0.872·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s + 1.35·35-s + 0.657·37-s − 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s − 0.539·55-s + 1.56·59-s + 1.28·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.17104944651594728900309356814, −6.97679761230185618004058366676, −6.25411171080538813643811013529, −5.34763163826409567254815856529, −4.30848928533479555552631006938, −3.82077695658162933193770448774, −3.12438273904855603476205552391, −2.49168947520422313000788800292, −1.12732292889285984272104789453, 0,
1.12732292889285984272104789453, 2.49168947520422313000788800292, 3.12438273904855603476205552391, 3.82077695658162933193770448774, 4.30848928533479555552631006938, 5.34763163826409567254815856529, 6.25411171080538813643811013529, 6.97679761230185618004058366676, 7.17104944651594728900309356814
