L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 7-s − 3·8-s + 9-s + 10-s + 12-s + 2·13-s + 14-s − 15-s − 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 21-s + 8·23-s + 3·24-s + 25-s + 2·26-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 + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.854127408\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.854127408\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.35996748906596, −15.65395438214757, −14.99132955833102, −14.53117026465047, −14.02429537723297, −13.42777061646823, −12.91929218404184, −12.48581408419415, −11.72783250278104, −11.33507533401237, −10.64296064523174, −9.865665790048647, −9.420743501559956, −8.864780280801617, −7.938798459835462, −7.503113197220087, −6.492663705753853, −5.946174895347315, −5.414025060139403, −4.873316741813236, −4.264399018678019, −3.304001328038768, −2.865086761099310, −1.429386043170342, −0.7953853860875148,
0.7953853860875148, 1.429386043170342, 2.865086761099310, 3.304001328038768, 4.264399018678019, 4.873316741813236, 5.414025060139403, 5.946174895347315, 6.492663705753853, 7.503113197220087, 7.938798459835462, 8.864780280801617, 9.420743501559956, 9.865665790048647, 10.64296064523174, 11.33507533401237, 11.72783250278104, 12.48581408419415, 12.91929218404184, 13.42777061646823, 14.02429537723297, 14.53117026465047, 14.99132955833102, 15.65395438214757, 16.35996748906596