| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 3·17-s − 18-s − 6·19-s + 20-s − 21-s + 2·22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918298041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918298041\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.35652506188294, −12.30026719584762, −11.59800278466475, −11.00812374655989, −10.83868414964196, −10.28282022442633, −10.05464214118558, −9.385001514576981, −9.038582481653637, −8.350260622783140, −8.099335613187831, −7.605571978725327, −7.045919759062916, −6.520135202173922, −6.015114874270576, −5.803529111386027, −5.022945060233129, −4.707648809221584, −4.043284367724024, −3.415667303644427, −2.734778012632047, −2.141033720105392, −1.759918919443075, −0.8954028508383895, −0.5249163316582929,
0.5249163316582929, 0.8954028508383895, 1.759918919443075, 2.141033720105392, 2.734778012632047, 3.415667303644427, 4.043284367724024, 4.707648809221584, 5.022945060233129, 5.803529111386027, 6.015114874270576, 6.520135202173922, 7.045919759062916, 7.605571978725327, 8.099335613187831, 8.350260622783140, 9.038582481653637, 9.385001514576981, 10.05464214118558, 10.28282022442633, 10.83868414964196, 11.00812374655989, 11.59800278466475, 12.30026719584762, 12.35652506188294
