| L(s) = 1 | − 2-s + 0.777·3-s + 4-s − 5-s − 0.777·6-s + 1.65·7-s − 8-s − 2.39·9-s + 10-s − 0.734·11-s + 0.777·12-s − 13-s − 1.65·14-s − 0.777·15-s + 16-s + 4.91·17-s + 2.39·18-s + 5.49·19-s − 20-s + 1.28·21-s + 0.734·22-s + 8.77·23-s − 0.777·24-s + 25-s + 26-s − 4.19·27-s + 1.65·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.448·3-s + 0.5·4-s − 0.447·5-s − 0.317·6-s + 0.625·7-s − 0.353·8-s − 0.798·9-s + 0.316·10-s − 0.221·11-s + 0.224·12-s − 0.277·13-s − 0.442·14-s − 0.200·15-s + 0.250·16-s + 1.19·17-s + 0.564·18-s + 1.26·19-s − 0.223·20-s + 0.280·21-s + 0.156·22-s + 1.83·23-s − 0.158·24-s + 0.200·25-s + 0.196·26-s − 0.807·27-s + 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472158791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472158791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.777T + 3T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 0.734T + 11T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 + 0.442T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 2.82T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.393706376496055707373953725049, −7.65197340274994913709936545144, −7.49076237977567704134432803893, −6.37003155080187482962150573174, −5.35485910226532077854104198740, −4.89525446239929148862074350929, −3.37256183227177484965895825550, −3.09146492593183804863243646316, −1.85706345697986317048306089433, −0.76595506878095868298627210928,
0.76595506878095868298627210928, 1.85706345697986317048306089433, 3.09146492593183804863243646316, 3.37256183227177484965895825550, 4.89525446239929148862074350929, 5.35485910226532077854104198740, 6.37003155080187482962150573174, 7.49076237977567704134432803893, 7.65197340274994913709936545144, 8.393706376496055707373953725049
