| L(s) = 1 | − 1.88·2-s + 1.55·4-s − 1.57·5-s − 2.47·7-s + 0.845·8-s + 2.97·10-s + 3.83·11-s − 3.18·13-s + 4.65·14-s − 4.69·16-s + 4.51·17-s + 4.54·19-s − 2.44·20-s − 7.22·22-s − 1.92·23-s − 2.51·25-s + 6.00·26-s − 3.83·28-s + 0.0587·29-s − 9.58·31-s + 7.15·32-s − 8.49·34-s + 3.89·35-s + 0.789·37-s − 8.56·38-s − 1.33·40-s + 2.07·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.775·4-s − 0.705·5-s − 0.934·7-s + 0.298·8-s + 0.939·10-s + 1.15·11-s − 0.883·13-s + 1.24·14-s − 1.17·16-s + 1.09·17-s + 1.04·19-s − 0.547·20-s − 1.54·22-s − 0.401·23-s − 0.502·25-s + 1.17·26-s − 0.724·28-s + 0.0109·29-s − 1.72·31-s + 1.26·32-s − 1.45·34-s + 0.658·35-s + 0.129·37-s − 1.38·38-s − 0.210·40-s + 0.324·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5219440589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5219440589\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 - 0.0587T + 29T^{2} \) |
| 31 | \( 1 + 9.58T + 31T^{2} \) |
| 37 | \( 1 - 0.789T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 0.844T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 2.97T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + 6.42T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−9.440668773715265489690457920842, −8.314760434658418845365595737978, −7.58555514265721617886941286280, −7.14175415448134590041552031271, −6.23154950600642281171492532411, −5.13355871632369959164809991789, −3.95972934251439923979662392231, −3.25115955239197695019260462690, −1.80957575256963323366859808270, −0.58186875671990591098887996610,
0.58186875671990591098887996610, 1.80957575256963323366859808270, 3.25115955239197695019260462690, 3.95972934251439923979662392231, 5.13355871632369959164809991789, 6.23154950600642281171492532411, 7.14175415448134590041552031271, 7.58555514265721617886941286280, 8.314760434658418845365595737978, 9.440668773715265489690457920842
