| L(s) = 1 | − 2.21·2-s − 2.91·3-s + 2.90·4-s − 0.513·5-s + 6.46·6-s + 5.13·7-s − 2.01·8-s + 5.51·9-s + 1.13·10-s + 1.76·11-s − 8.48·12-s + 2.99·13-s − 11.3·14-s + 1.49·15-s − 1.35·16-s + 17-s − 12.2·18-s + 4.90·19-s − 1.49·20-s − 14.9·21-s − 3.91·22-s − 5.53·23-s + 5.88·24-s − 4.73·25-s − 6.64·26-s − 7.32·27-s + 14.9·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 1.68·3-s + 1.45·4-s − 0.229·5-s + 2.63·6-s + 1.94·7-s − 0.712·8-s + 1.83·9-s + 0.360·10-s + 0.532·11-s − 2.45·12-s + 0.831·13-s − 3.04·14-s + 0.387·15-s − 0.338·16-s + 0.242·17-s − 2.87·18-s + 1.12·19-s − 0.334·20-s − 3.26·21-s − 0.834·22-s − 1.15·23-s + 1.20·24-s − 0.947·25-s − 1.30·26-s − 1.41·27-s + 2.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5237628702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5237628702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 + 2.91T + 3T^{2} \) |
| 5 | \( 1 + 0.513T + 5T^{2} \) |
| 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 1.76T + 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 - 0.605T + 37T^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 3.77T + 59T^{2} \) |
| 61 | \( 1 - 8.99T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 4.75T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 0.413T + 83T^{2} \) |
| 89 | \( 1 + 0.308T + 89T^{2} \) |
| 97 | \( 1 - 3.57T + 97T^{2} \) |
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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
−10.38593550495623247667989186640, −9.804628084876668638256397012117, −8.467555565566726542230976693763, −7.985921640653762320815546695169, −7.07565201950153667046487739823, −6.11128397471583054553714157034, −5.13870042214594515691286862194, −4.22325015399219898839391886181, −1.70141211825815803925597649775, −0.919983366968794242497869123708,
0.919983366968794242497869123708, 1.70141211825815803925597649775, 4.22325015399219898839391886181, 5.13870042214594515691286862194, 6.11128397471583054553714157034, 7.07565201950153667046487739823, 7.985921640653762320815546695169, 8.467555565566726542230976693763, 9.804628084876668638256397012117, 10.38593550495623247667989186640
