| L(s) = 1 | − 2.12·2-s − 3.49·4-s + 4.52·5-s + 14.7·7-s + 24.3·8-s − 9.60·10-s − 61.2·11-s − 8.40·13-s − 31.3·14-s − 23.7·16-s + 127.·17-s − 43.6·19-s − 15.8·20-s + 130.·22-s − 165.·23-s − 104.·25-s + 17.8·26-s − 51.7·28-s + 212.·29-s + 213.·31-s − 144.·32-s − 270.·34-s + 66.9·35-s + 56.0·37-s + 92.6·38-s + 110.·40-s − 328.·41-s + ⋯ |
| L(s) = 1 | − 0.750·2-s − 0.437·4-s + 0.405·5-s + 0.798·7-s + 1.07·8-s − 0.303·10-s − 1.67·11-s − 0.179·13-s − 0.598·14-s − 0.371·16-s + 1.82·17-s − 0.527·19-s − 0.177·20-s + 1.26·22-s − 1.50·23-s − 0.835·25-s + 0.134·26-s − 0.349·28-s + 1.36·29-s + 1.23·31-s − 0.799·32-s − 1.36·34-s + 0.323·35-s + 0.248·37-s + 0.395·38-s + 0.436·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.122224763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122224763\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 2.12T + 8T^{2} \) |
| 5 | \( 1 - 4.52T + 125T^{2} \) |
| 7 | \( 1 - 14.7T + 343T^{2} \) |
| 11 | \( 1 + 61.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.40T + 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 56.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 259.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 807.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 437.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 641.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 417.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 236.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 67.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 13.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 471.T + 9.12e5T^{2} \) |
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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
−8.440010675096965384563612079762, −8.031302450402769784691780983433, −7.71196794182836222155814527640, −6.39507460579423753215841340075, −5.33447740668933622287907137003, −4.95535652167281292071421614817, −3.86911552687122484185571670795, −2.59156491699850186513449797374, −1.64414920921264031282396931972, −0.55447870068562057628836695960,
0.55447870068562057628836695960, 1.64414920921264031282396931972, 2.59156491699850186513449797374, 3.86911552687122484185571670795, 4.95535652167281292071421614817, 5.33447740668933622287907137003, 6.39507460579423753215841340075, 7.71196794182836222155814527640, 8.031302450402769784691780983433, 8.440010675096965384563612079762
