| L(s) = 1 | − 1.48·2-s + 1.67·3-s + 0.193·4-s + 5-s − 2.48·6-s + 7-s + 2.67·8-s − 0.193·9-s − 1.48·10-s + 11-s + 0.324·12-s + 1.67·13-s − 1.48·14-s + 1.67·15-s − 4.35·16-s + 0.324·17-s + 0.287·18-s + 3.61·19-s + 0.193·20-s + 1.67·21-s − 1.48·22-s + 0.806·23-s + 4.48·24-s + 25-s − 2.48·26-s − 5.35·27-s + 0.193·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.967·3-s + 0.0969·4-s + 0.447·5-s − 1.01·6-s + 0.377·7-s + 0.945·8-s − 0.0646·9-s − 0.468·10-s + 0.301·11-s + 0.0937·12-s + 0.464·13-s − 0.395·14-s + 0.432·15-s − 1.08·16-s + 0.0787·17-s + 0.0677·18-s + 0.828·19-s + 0.0433·20-s + 0.365·21-s − 0.315·22-s + 0.168·23-s + 0.914·24-s + 0.200·25-s − 0.486·26-s − 1.02·27-s + 0.0366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160798095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160798095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 - 0.324T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 + 7.76T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 4.15T + 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 2.15T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 6.05T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−11.00710993926697472284710998286, −10.14204919104358705082524055188, −9.274784710234536703729546416823, −8.689966626180650695555070809467, −7.983207999303971417808726656245, −7.01734653552297031788491803960, −5.57578652022419402982411137477, −4.22445257389763833690178776697, −2.78547252804522878341759041264, −1.35360706849061843707222993155,
1.35360706849061843707222993155, 2.78547252804522878341759041264, 4.22445257389763833690178776697, 5.57578652022419402982411137477, 7.01734653552297031788491803960, 7.983207999303971417808726656245, 8.689966626180650695555070809467, 9.274784710234536703729546416823, 10.14204919104358705082524055188, 11.00710993926697472284710998286
