| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s − 6·11-s + 3·13-s − 8·17-s − 6·21-s + 23-s − 5·25-s + 9·27-s − 3·29-s + 7·31-s − 18·33-s − 8·37-s + 9·39-s + 7·41-s − 6·43-s − 9·47-s − 3·49-s − 24·51-s − 12·53-s + 12·59-s − 4·61-s − 12·63-s + 2·67-s + 3·69-s − 5·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s − 1.80·11-s + 0.832·13-s − 1.94·17-s − 1.30·21-s + 0.208·23-s − 25-s + 1.73·27-s − 0.557·29-s + 1.25·31-s − 3.13·33-s − 1.31·37-s + 1.44·39-s + 1.09·41-s − 0.914·43-s − 1.31·47-s − 3/7·49-s − 3.36·51-s − 1.64·53-s + 1.56·59-s − 0.512·61-s − 1.51·63-s + 0.244·67-s + 0.361·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213118876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213118876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.01991555149106, −13.55307583608730, −13.32974244580160, −12.95628910966310, −12.59838358704611, −11.57420893840401, −11.12784557345305, −10.45116444856347, −10.04099350208811, −9.566669333165708, −9.001229597151893, −8.437010086068214, −8.255277510334020, −7.567164169900592, −7.118188245061283, −6.414323322038216, −5.976725615299370, −4.960660178942674, −4.563076260651667, −3.802287375055052, −3.218403309001539, −2.864195315286833, −2.102571451919013, −1.781245320745224, −0.4055015594690182,
0.4055015594690182, 1.781245320745224, 2.102571451919013, 2.864195315286833, 3.218403309001539, 3.802287375055052, 4.563076260651667, 4.960660178942674, 5.976725615299370, 6.414323322038216, 7.118188245061283, 7.567164169900592, 8.255277510334020, 8.437010086068214, 9.001229597151893, 9.566669333165708, 10.04099350208811, 10.45116444856347, 11.12784557345305, 11.57420893840401, 12.59838358704611, 12.95628910966310, 13.32974244580160, 13.55307583608730, 14.01991555149106
