| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 11-s + 12-s + 5·13-s − 15-s + 16-s + 17-s − 18-s + 8·19-s − 20-s + 22-s − 6·23-s − 24-s − 4·25-s − 5·26-s + 27-s − 4·29-s + 30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.38·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s − 0.980·26-s + 0.192·27-s − 0.742·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699384214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699384214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.179494298748499458005431761875, −7.62109987737726972185862957933, −7.24579546300677593863242971552, −5.96545609487344741051017416089, −5.67332621373794895233557698762, −4.25055603436508733716576977643, −3.64490974682243201539180760032, −2.83571486026884702125696225150, −1.78107708587430495090295524181, −0.795074797552514890210776216705,
0.795074797552514890210776216705, 1.78107708587430495090295524181, 2.83571486026884702125696225150, 3.64490974682243201539180760032, 4.25055603436508733716576977643, 5.67332621373794895233557698762, 5.96545609487344741051017416089, 7.24579546300677593863242971552, 7.62109987737726972185862957933, 8.179494298748499458005431761875
