| L(s) = 1 | − 2-s + 0.307·3-s + 4-s − 2.84·5-s − 0.307·6-s − 3.84·7-s − 8-s − 2.90·9-s + 2.84·10-s + 2.26·11-s + 0.307·12-s + 6.67·13-s + 3.84·14-s − 0.877·15-s + 16-s + 4.39·17-s + 2.90·18-s + 19-s − 2.84·20-s − 1.18·21-s − 2.26·22-s + 5.89·23-s − 0.307·24-s + 3.11·25-s − 6.67·26-s − 1.81·27-s − 3.84·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.177·3-s + 0.5·4-s − 1.27·5-s − 0.125·6-s − 1.45·7-s − 0.353·8-s − 0.968·9-s + 0.901·10-s + 0.683·11-s + 0.0888·12-s + 1.85·13-s + 1.02·14-s − 0.226·15-s + 0.250·16-s + 1.06·17-s + 0.684·18-s + 0.229·19-s − 0.637·20-s − 0.258·21-s − 0.483·22-s + 1.22·23-s − 0.0628·24-s + 0.623·25-s − 1.30·26-s − 0.349·27-s − 0.726·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7193726286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7193726286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.307T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 6.67T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 23 | \( 1 - 5.89T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 + 3.94T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.67T + 59T^{2} \) |
| 61 | \( 1 - 6.44T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 0.0439T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 8.26T + 97T^{2} \) |
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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
−8.047405541112643028670412912447, −7.19058263416009099437443710314, −6.54600332908166075238657177925, −6.03289196442984479786431011920, −5.16911825885598457605786090606, −3.74170106532899846049356027491, −3.48508284036617927921204575782, −3.01355522649473555162650848514, −1.47887257724623946341263656111, −0.48366810071751240234521009157,
0.48366810071751240234521009157, 1.47887257724623946341263656111, 3.01355522649473555162650848514, 3.48508284036617927921204575782, 3.74170106532899846049356027491, 5.16911825885598457605786090606, 6.03289196442984479786431011920, 6.54600332908166075238657177925, 7.19058263416009099437443710314, 8.047405541112643028670412912447
