| L(s) = 1 | + 2-s − 3-s + 4-s + 3.89·5-s − 6-s + 2·7-s + 8-s + 9-s + 3.89·10-s − 2.35·11-s − 12-s + 1.54·13-s + 2·14-s − 3.89·15-s + 16-s − 1.08·17-s + 18-s + 19-s + 3.89·20-s − 2·21-s − 2.35·22-s − 0.459·23-s − 24-s + 10.1·25-s + 1.54·26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.74·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.333·9-s + 1.23·10-s − 0.709·11-s − 0.288·12-s + 0.427·13-s + 0.534·14-s − 1.00·15-s + 0.250·16-s − 0.262·17-s + 0.235·18-s + 0.229·19-s + 0.870·20-s − 0.436·21-s − 0.501·22-s − 0.0957·23-s − 0.204·24-s + 2.03·25-s + 0.302·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.274319115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.274319115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 23 | \( 1 + 0.459T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 + 3.78T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 0.918T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 5.78T + 97T^{2} \) |
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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
−8.036303688447286957174276756322, −7.06351736618180196002747977358, −6.42433294871180807881056010617, −5.83966128394009698249240664422, −5.15006097736211135595157985619, −4.87464669298127865868264617480, −3.72834507213728064810789060110, −2.59017372667119811377463218590, −1.97343870997856565161290734961, −1.07441049948081605451252133932,
1.07441049948081605451252133932, 1.97343870997856565161290734961, 2.59017372667119811377463218590, 3.72834507213728064810789060110, 4.87464669298127865868264617480, 5.15006097736211135595157985619, 5.83966128394009698249240664422, 6.42433294871180807881056010617, 7.06351736618180196002747977358, 8.036303688447286957174276756322
