L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s + 21-s + 22-s − 23-s − 24-s − 26-s + 27-s + 28-s − 29-s + 31-s − 32-s − 33-s − 34-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s + 21-s + 22-s − 23-s − 24-s − 26-s + 27-s + 28-s − 29-s + 31-s − 32-s − 33-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.178797996\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178797996\) |
\(L(1)\) |
\(\approx\) |
\(1.050623227\) |
\(L(1)\) |
\(\approx\) |
\(1.050623227\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.650081450523990323103382270908, −25.95487097792580062433321904997, −25.24244460715117427514097224097, −24.20338157764452992388433105640, −23.495536444298538892381093265923, −21.441769784801916958895408899801, −20.887352837830433919899423736621, −20.2491204078381543444947392479, −18.93455295003832081472092399512, −18.47203535670177198485308339104, −17.43774452291833594336007339150, −16.16176775378698397285670808550, −15.321245939053144792288746344625, −14.43421659974855760358259968221, −13.255727723864375319831488514872, −11.95919263049457276358123342659, −10.7088831302949322050595593809, −9.97311100371886289159348175316, −8.52839994915370115591979491986, −8.192765739144116330406084965851, −7.14147924049611037154518982653, −5.60893181841362294221561065449, −3.885211036652835593591011229575, −2.50253086009588512448210890372, −1.46732342903323083553918841160,
1.46732342903323083553918841160, 2.50253086009588512448210890372, 3.885211036652835593591011229575, 5.60893181841362294221561065449, 7.14147924049611037154518982653, 8.192765739144116330406084965851, 8.52839994915370115591979491986, 9.97311100371886289159348175316, 10.7088831302949322050595593809, 11.95919263049457276358123342659, 13.255727723864375319831488514872, 14.43421659974855760358259968221, 15.321245939053144792288746344625, 16.16176775378698397285670808550, 17.43774452291833594336007339150, 18.47203535670177198485308339104, 18.93455295003832081472092399512, 20.2491204078381543444947392479, 20.887352837830433919899423736621, 21.441769784801916958895408899801, 23.495536444298538892381093265923, 24.20338157764452992388433105640, 25.24244460715117427514097224097, 25.95487097792580062433321904997, 26.650081450523990323103382270908