| L(s) = 1 | + 2.61·2-s + 4.84·4-s + 2.80·5-s + 3.73·7-s + 7.43·8-s + 7.34·10-s − 6.02·11-s − 0.651·13-s + 9.77·14-s + 9.75·16-s + 1.63·17-s − 6.09·19-s + 13.6·20-s − 15.7·22-s + 1.60·23-s + 2.89·25-s − 1.70·26-s + 18.0·28-s + 2.35·29-s − 6.90·31-s + 10.6·32-s + 4.27·34-s + 10.4·35-s − 10.0·37-s − 15.9·38-s + 20.8·40-s + 8.60·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.42·4-s + 1.25·5-s + 1.41·7-s + 2.62·8-s + 2.32·10-s − 1.81·11-s − 0.180·13-s + 2.61·14-s + 2.43·16-s + 0.396·17-s − 1.39·19-s + 3.04·20-s − 3.35·22-s + 0.334·23-s + 0.578·25-s − 0.334·26-s + 3.41·28-s + 0.437·29-s − 1.24·31-s + 1.88·32-s + 0.733·34-s + 1.77·35-s − 1.65·37-s − 2.58·38-s + 3.30·40-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.263217489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.263217489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 + 6.02T + 11T^{2} \) |
| 13 | \( 1 + 0.651T + 13T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 - 8.78T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 + 9.50T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 9.36T + 71T^{2} \) |
| 73 | \( 1 - 8.06T + 73T^{2} \) |
| 79 | \( 1 + 4.58T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−9.049667627225675620694523886553, −7.927599736221076412495241319876, −7.41751035105500597061290748848, −6.25607897126617317093628055751, −5.70083390916800414695825110055, −4.95685551118193955932580079854, −4.64868877884214770952804135107, −3.27670712595879138227748844378, −2.25710474672254949538887910736, −1.83923177796702803955643412343,
1.83923177796702803955643412343, 2.25710474672254949538887910736, 3.27670712595879138227748844378, 4.64868877884214770952804135107, 4.95685551118193955932580079854, 5.70083390916800414695825110055, 6.25607897126617317093628055751, 7.41751035105500597061290748848, 7.927599736221076412495241319876, 9.049667627225675620694523886553
