| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s − 11-s + 12-s + 4·13-s − 14-s − 15-s − 16-s − 18-s − 20-s − 21-s + 22-s + 5·23-s − 3·24-s − 4·25-s − 4·26-s − 27-s − 28-s + 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.04·23-s − 0.612·24-s − 4/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498918076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498918076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.96735845098674, −13.37504247530611, −13.14452288282980, −12.39907861431142, −11.97431328322381, −11.20952666336110, −10.93872186096587, −10.42475714072824, −9.936677727681340, −9.518412923679581, −8.882335794239186, −8.394742803039572, −8.082468753783215, −7.407630304500282, −6.721449852897348, −6.356035016623078, −5.573140762331094, −5.123865576330860, −4.704571970044273, −3.950742180916432, −3.443117445086634, −2.482664123623059, −1.734999271212115, −1.099068492281069, −0.5557026707195660,
0.5557026707195660, 1.099068492281069, 1.734999271212115, 2.482664123623059, 3.443117445086634, 3.950742180916432, 4.704571970044273, 5.123865576330860, 5.573140762331094, 6.356035016623078, 6.721449852897348, 7.407630304500282, 8.082468753783215, 8.394742803039572, 8.882335794239186, 9.518412923679581, 9.936677727681340, 10.42475714072824, 10.93872186096587, 11.20952666336110, 11.97431328322381, 12.39907861431142, 13.14452288282980, 13.37504247530611, 13.96735845098674
