| L(s) = 1 | + 0.835·2-s − 2.05·3-s − 1.30·4-s + 3.71·5-s − 1.71·6-s − 2.16·7-s − 2.75·8-s + 1.20·9-s + 3.09·10-s − 3.85·11-s + 2.67·12-s + 13-s − 1.81·14-s − 7.60·15-s + 0.301·16-s + 7.10·17-s + 1.00·18-s + 0.0250·19-s − 4.83·20-s + 4.44·21-s − 3.22·22-s + 1.83·23-s + 5.65·24-s + 8.76·25-s + 0.835·26-s + 3.68·27-s + 2.82·28-s + ⋯ |
| L(s) = 1 | + 0.590·2-s − 1.18·3-s − 0.651·4-s + 1.65·5-s − 0.699·6-s − 0.820·7-s − 0.975·8-s + 0.401·9-s + 0.979·10-s − 1.16·11-s + 0.771·12-s + 0.277·13-s − 0.484·14-s − 1.96·15-s + 0.0754·16-s + 1.72·17-s + 0.237·18-s + 0.00574·19-s − 1.08·20-s + 0.970·21-s − 0.687·22-s + 0.383·23-s + 1.15·24-s + 1.75·25-s + 0.163·26-s + 0.708·27-s + 0.534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296758339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296758339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.835T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 - 0.0250T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 - 9.81T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 5.53T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 0.0829T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 7.73T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−9.661261927248809512271457910195, −9.204512836453583827834099244126, −7.953640173959477264379736040940, −6.69493282820092190262048811862, −5.85317466985553739442817363773, −5.54802577995344905705886628379, −5.01605521541174805862533376340, −3.55533683677727385764399944630, −2.54338286751953737303551061847, −0.806741873299991159293438854364,
0.806741873299991159293438854364, 2.54338286751953737303551061847, 3.55533683677727385764399944630, 5.01605521541174805862533376340, 5.54802577995344905705886628379, 5.85317466985553739442817363773, 6.69493282820092190262048811862, 7.953640173959477264379736040940, 9.204512836453583827834099244126, 9.661261927248809512271457910195
