| L(s) = 1 | − 2.56·2-s + 3-s + 4.56·4-s − 1.56·5-s − 2.56·6-s + 7-s − 6.56·8-s + 9-s + 4·10-s − 2.43·11-s + 4.56·12-s + 2·13-s − 2.56·14-s − 1.56·15-s + 7.68·16-s − 2·17-s − 2.56·18-s − 2.43·19-s − 7.12·20-s + 21-s + 6.24·22-s − 3.12·23-s − 6.56·24-s − 2.56·25-s − 5.12·26-s + 27-s + 4.56·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.698·5-s − 1.04·6-s + 0.377·7-s − 2.31·8-s + 0.333·9-s + 1.26·10-s − 0.735·11-s + 1.31·12-s + 0.554·13-s − 0.684·14-s − 0.403·15-s + 1.92·16-s − 0.485·17-s − 0.603·18-s − 0.559·19-s − 1.59·20-s + 0.218·21-s + 1.33·22-s − 0.651·23-s − 1.33·24-s − 0.512·25-s − 1.00·26-s + 0.192·27-s + 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 0.438T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−7.76737368448892512548724295295, −7.21231082826054168397471167754, −6.46506587496011153327561573421, −5.68309803314568383917737269824, −4.47702651977205830771714179261, −3.73664676378171984159024244614, −2.64633050743825331430755292951, −2.10801075472669542844270756903, −1.06348592998466505165400905169, 0,
1.06348592998466505165400905169, 2.10801075472669542844270756903, 2.64633050743825331430755292951, 3.73664676378171984159024244614, 4.47702651977205830771714179261, 5.68309803314568383917737269824, 6.46506587496011153327561573421, 7.21231082826054168397471167754, 7.76737368448892512548724295295
