| L(s) = 1 | + 1.14·3-s + 3.83·5-s + 7-s − 1.68·9-s + 4.68·11-s − 5.53·13-s + 4.39·15-s + 0.292·17-s + 5.14·19-s + 1.14·21-s + 4.97·23-s + 9.68·25-s − 5.37·27-s − 4.29·29-s − 7.66·31-s + 5.37·33-s + 3.83·35-s − 9.66·37-s − 6.35·39-s + 3.70·41-s − 5.27·43-s − 6.46·45-s + 2.29·47-s + 49-s + 0.335·51-s + 2·53-s + 17.9·55-s + ⋯ |
| L(s) = 1 | + 0.661·3-s + 1.71·5-s + 0.377·7-s − 0.561·9-s + 1.41·11-s − 1.53·13-s + 1.13·15-s + 0.0709·17-s + 1.18·19-s + 0.250·21-s + 1.03·23-s + 1.93·25-s − 1.03·27-s − 0.797·29-s − 1.37·31-s + 0.935·33-s + 0.647·35-s − 1.58·37-s − 1.01·39-s + 0.578·41-s − 0.803·43-s − 0.963·45-s + 0.334·47-s + 0.142·49-s + 0.0469·51-s + 0.274·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.645602674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645602674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 - 5.14T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 + 9.66T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 9.93T + 59T^{2} \) |
| 61 | \( 1 + 4.16T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.763606516894157159270186011621, −9.301132719933304070962618535490, −8.816806664379075350120375627003, −7.47755627218437135564505800010, −6.76059070168984139038210218522, −5.59178587458064865644905131960, −5.08954760384519037715211600453, −3.52309029022328674685815819149, −2.44564488916745653411691916876, −1.54957319981715009338241219472,
1.54957319981715009338241219472, 2.44564488916745653411691916876, 3.52309029022328674685815819149, 5.08954760384519037715211600453, 5.59178587458064865644905131960, 6.76059070168984139038210218522, 7.47755627218437135564505800010, 8.816806664379075350120375627003, 9.301132719933304070962618535490, 9.763606516894157159270186011621
