| L(s) = 1 | + 0.300·3-s − 5-s + 7-s − 2.90·9-s − 11-s + 3.93·13-s − 0.300·15-s − 7.53·17-s − 5.23·19-s + 0.300·21-s + 4.09·23-s + 25-s − 1.77·27-s − 3.23·29-s + 1.35·31-s − 0.300·33-s − 35-s + 5.49·37-s + 1.17·39-s + 3.43·41-s − 9.12·43-s + 2.90·45-s + 6.13·47-s + 49-s − 2.26·51-s + 3.67·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.173·3-s − 0.447·5-s + 0.377·7-s − 0.969·9-s − 0.301·11-s + 1.09·13-s − 0.0774·15-s − 1.82·17-s − 1.20·19-s + 0.0654·21-s + 0.853·23-s + 0.200·25-s − 0.341·27-s − 0.600·29-s + 0.242·31-s − 0.0522·33-s − 0.169·35-s + 0.903·37-s + 0.188·39-s + 0.537·41-s − 1.39·43-s + 0.433·45-s + 0.894·47-s + 0.142·49-s − 0.316·51-s + 0.505·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427435189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427435189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.300T + 3T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 5.49T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 - 7.79T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.337111373897704051365522317698, −7.37098100181150985470609447119, −6.61802749322198999446009052019, −6.00822607594885401797093037998, −5.15731915875154564177899307212, −4.34045437069021639400039929717, −3.71982198020740735973288903032, −2.69627807517213194752479553286, −2.00913961889317382786911489561, −0.59741954148587353718148734537,
0.59741954148587353718148734537, 2.00913961889317382786911489561, 2.69627807517213194752479553286, 3.71982198020740735973288903032, 4.34045437069021639400039929717, 5.15731915875154564177899307212, 6.00822607594885401797093037998, 6.61802749322198999446009052019, 7.37098100181150985470609447119, 8.337111373897704051365522317698
