| L(s) = 1 | + 2·2-s − 9·3-s + 4·4-s + 20·5-s − 18·6-s − 2·7-s + 8·8-s + 54·9-s + 40·10-s + 52·11-s − 36·12-s + 43·13-s − 4·14-s − 180·15-s + 16·16-s + 50·17-s + 108·18-s + 74·19-s + 80·20-s + 18·21-s + 104·22-s − 72·24-s + 275·25-s + 86·26-s − 243·27-s − 8·28-s − 7·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.78·5-s − 1.22·6-s − 0.107·7-s + 0.353·8-s + 2·9-s + 1.26·10-s + 1.42·11-s − 0.866·12-s + 0.917·13-s − 0.0763·14-s − 3.09·15-s + 1/4·16-s + 0.713·17-s + 1.41·18-s + 0.893·19-s + 0.894·20-s + 0.187·21-s + 1.00·22-s − 0.612·24-s + 11/5·25-s + 0.648·26-s − 1.73·27-s − 0.0539·28-s − 0.0448·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.368637408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368637408\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 273 T + p^{3} T^{2} \) |
| 37 | \( 1 - 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 75 T + p^{3} T^{2} \) |
| 53 | \( 1 + 86 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 681 T + p^{3} T^{2} \) |
| 79 | \( 1 + 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 342 T + p^{3} T^{2} \) |
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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
−9.690391745609265231296084089470, −9.107673982060656499193684015517, −7.32023413802923643502471607137, −6.47790012349066323044157039967, −5.94940490753389475679828375647, −5.54237146862631509075664156284, −4.60017578403263171197325565748, −3.41082280492892568525898940268, −1.69543567478314322126139462052, −1.08828736899378739737913408212,
1.08828736899378739737913408212, 1.69543567478314322126139462052, 3.41082280492892568525898940268, 4.60017578403263171197325565748, 5.54237146862631509075664156284, 5.94940490753389475679828375647, 6.47790012349066323044157039967, 7.32023413802923643502471607137, 9.107673982060656499193684015517, 9.690391745609265231296084089470
