| L(s) = 1 | + 1.48·2-s − 3-s + 0.193·4-s − 1.48·6-s − 2.48·7-s − 2.67·8-s + 9-s + 2·11-s − 0.193·12-s + 5.83·13-s − 3.67·14-s − 4.35·16-s − 3.76·17-s + 1.48·18-s + 2.48·21-s + 2.96·22-s − 0.806·23-s + 2.67·24-s + 8.63·26-s − 27-s − 0.481·28-s + 4.06·29-s + 31-s − 1.09·32-s − 2·33-s − 5.58·34-s + 0.193·36-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.577·3-s + 0.0969·4-s − 0.604·6-s − 0.937·7-s − 0.945·8-s + 0.333·9-s + 0.603·11-s − 0.0559·12-s + 1.61·13-s − 0.982·14-s − 1.08·16-s − 0.913·17-s + 0.349·18-s + 0.541·21-s + 0.631·22-s − 0.168·23-s + 0.546·24-s + 1.69·26-s − 0.192·27-s − 0.0909·28-s + 0.754·29-s + 0.179·31-s − 0.193·32-s − 0.348·33-s − 0.957·34-s + 0.0323·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.942261301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942261301\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 0.806T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 0.0303T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.989966622851998180001648346021, −8.407935084563040393962698948800, −6.95088041557994303712750740934, −6.43030420985429735388412819092, −5.88500635607696213266206251870, −5.05892755735292893382154273910, −3.94064044292520968842852935283, −3.72094691786700280371159092801, −2.44580825382651754409294717805, −0.792530644872565892630190769112,
0.792530644872565892630190769112, 2.44580825382651754409294717805, 3.72094691786700280371159092801, 3.94064044292520968842852935283, 5.05892755735292893382154273910, 5.88500635607696213266206251870, 6.43030420985429735388412819092, 6.95088041557994303712750740934, 8.407935084563040393962698948800, 8.989966622851998180001648346021
