| L(s) = 1 | + 3-s − 2·7-s + 9-s + 4·11-s + 6·13-s + 6·17-s − 2·21-s − 4·23-s − 5·25-s + 27-s + 4·29-s − 10·31-s + 4·33-s + 2·37-s + 6·39-s − 2·41-s − 8·43-s + 12·47-s − 3·49-s + 6·51-s − 12·53-s + 4·59-s + 2·61-s − 2·63-s − 4·67-s − 4·69-s + 4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.436·21-s − 0.834·23-s − 25-s + 0.192·27-s + 0.742·29-s − 1.79·31-s + 0.696·33-s + 0.328·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s + 0.840·51-s − 1.64·53-s + 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.488·67-s − 0.481·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696119693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696119693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.40154081034543017478165859065, −10.24506957450820273783788363514, −9.464663439718228165986962411065, −8.658903060264415544055202217864, −7.70036027517747780731408099423, −6.52243750033071865852983380426, −5.75568326693151721019291253980, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −1.50413098873776121963751941726,
1.50413098873776121963751941726, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 5.75568326693151721019291253980, 6.52243750033071865852983380426, 7.70036027517747780731408099423, 8.658903060264415544055202217864, 9.464663439718228165986962411065, 10.24506957450820273783788363514, 11.40154081034543017478165859065
