| L(s) = 1 | + 2.04·3-s + 0.222·5-s + 5.15·7-s + 1.17·9-s + 2.64·11-s + 3.46·13-s + 0.455·15-s − 5.28·17-s + 0.248·19-s + 10.5·21-s − 0.439·23-s − 4.95·25-s − 3.72·27-s + 2.22·29-s − 2.80·31-s + 5.40·33-s + 1.14·35-s + 3.68·37-s + 7.09·39-s + 10.7·41-s − 12.6·43-s + 0.263·45-s + 7.58·47-s + 19.5·49-s − 10.7·51-s + 0.431·53-s + 0.589·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.0997·5-s + 1.94·7-s + 0.393·9-s + 0.797·11-s + 0.962·13-s + 0.117·15-s − 1.28·17-s + 0.0569·19-s + 2.30·21-s − 0.0916·23-s − 0.990·25-s − 0.716·27-s + 0.412·29-s − 0.503·31-s + 0.940·33-s + 0.194·35-s + 0.605·37-s + 1.13·39-s + 1.67·41-s − 1.92·43-s + 0.0392·45-s + 1.10·47-s + 2.79·49-s − 1.51·51-s + 0.0592·53-s + 0.0794·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.577827087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.577827087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 0.222T + 5T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 - 0.248T + 19T^{2} \) |
| 23 | \( 1 + 0.439T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 3.68T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 0.431T + 53T^{2} \) |
| 59 | \( 1 - 9.93T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 - 4.00T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 4.95T + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−8.108505134136072999542974620348, −7.38346985438540266618159322981, −6.52628736126422071928410302771, −5.72218959538385474740692240845, −4.89043847319569010827481655535, −4.08859461286104981762248630046, −3.70235926073084915034610466107, −2.39390729330606406052249206802, −1.96974246601333607386302130474, −1.08019998661371371177856459912,
1.08019998661371371177856459912, 1.96974246601333607386302130474, 2.39390729330606406052249206802, 3.70235926073084915034610466107, 4.08859461286104981762248630046, 4.89043847319569010827481655535, 5.72218959538385474740692240845, 6.52628736126422071928410302771, 7.38346985438540266618159322981, 8.108505134136072999542974620348
