| L(s) = 1 | + 2-s − 0.383·3-s + 4-s + 4.15·5-s − 0.383·6-s + 4.87·7-s + 8-s − 2.85·9-s + 4.15·10-s − 1.62·11-s − 0.383·12-s − 2.16·13-s + 4.87·14-s − 1.59·15-s + 16-s − 6.78·17-s − 2.85·18-s − 8.61·19-s + 4.15·20-s − 1.87·21-s − 1.62·22-s + 2.76·23-s − 0.383·24-s + 12.2·25-s − 2.16·26-s + 2.24·27-s + 4.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.221·3-s + 0.5·4-s + 1.85·5-s − 0.156·6-s + 1.84·7-s + 0.353·8-s − 0.950·9-s + 1.31·10-s − 0.491·11-s − 0.110·12-s − 0.600·13-s + 1.30·14-s − 0.411·15-s + 0.250·16-s − 1.64·17-s − 0.672·18-s − 1.97·19-s + 0.928·20-s − 0.408·21-s − 0.347·22-s + 0.577·23-s − 0.0783·24-s + 2.45·25-s − 0.424·26-s + 0.432·27-s + 0.921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.777535167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.777535167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 + 0.383T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 + 8.61T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 2.10T + 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 1.37T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 - 3.31T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−10.96853357505313360109238534113, −10.25473670523262595464780038854, −8.906953193488809951513603862854, −8.322390217729559574222891389775, −6.84614362028388658680482440779, −6.02621463315702238041053301608, −5.11012022115790680600725507479, −4.64347787242201747222359179862, −2.44668178923068732859400018018, −1.94268810267652616400949731135,
1.94268810267652616400949731135, 2.44668178923068732859400018018, 4.64347787242201747222359179862, 5.11012022115790680600725507479, 6.02621463315702238041053301608, 6.84614362028388658680482440779, 8.322390217729559574222891389775, 8.906953193488809951513603862854, 10.25473670523262595464780038854, 10.96853357505313360109238534113
