| L(s) = 1 | − 0.305·2-s − 3-s − 1.90·4-s − 0.276·5-s + 0.305·6-s + 7-s + 1.19·8-s + 9-s + 0.0845·10-s − 5.49·11-s + 1.90·12-s + 5.85·13-s − 0.305·14-s + 0.276·15-s + 3.44·16-s + 1.15·17-s − 0.305·18-s − 6.10·19-s + 0.527·20-s − 21-s + 1.68·22-s + 4.90·23-s − 1.19·24-s − 4.92·25-s − 1.78·26-s − 27-s − 1.90·28-s + ⋯ |
| L(s) = 1 | − 0.216·2-s − 0.577·3-s − 0.953·4-s − 0.123·5-s + 0.124·6-s + 0.377·7-s + 0.422·8-s + 0.333·9-s + 0.0267·10-s − 1.65·11-s + 0.550·12-s + 1.62·13-s − 0.0817·14-s + 0.0713·15-s + 0.861·16-s + 0.279·17-s − 0.0720·18-s − 1.40·19-s + 0.117·20-s − 0.218·21-s + 0.358·22-s + 1.02·23-s − 0.243·24-s − 0.984·25-s − 0.350·26-s − 0.192·27-s − 0.360·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7841173729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7841173729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.305T + 2T^{2} \) |
| 5 | \( 1 + 0.276T + 5T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 9.23T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 6.75T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.36T + 61T^{2} \) |
| 67 | \( 1 - 3.74T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 0.769T + 73T^{2} \) |
| 79 | \( 1 + 0.396T + 79T^{2} \) |
| 83 | \( 1 + 2.27T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−8.559708772014535439341353923095, −8.331102968948135724600194548438, −7.51942433838880145872689459287, −6.45992301706965762887576444868, −5.58567895748606515787054747226, −5.03769652702255148342748592017, −4.18657772225276913305668607177, −3.33214673607384055944862923959, −1.87573168663846264414950532925, −0.59643922799407332513135805917,
0.59643922799407332513135805917, 1.87573168663846264414950532925, 3.33214673607384055944862923959, 4.18657772225276913305668607177, 5.03769652702255148342748592017, 5.58567895748606515787054747226, 6.45992301706965762887576444868, 7.51942433838880145872689459287, 8.331102968948135724600194548438, 8.559708772014535439341353923095
