| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 12-s + 13-s − 2·14-s − 15-s + 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 2·21-s − 3·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 4·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.78705540282558, −14.26863342255615, −13.80726118952655, −13.18924047285203, −12.72063932410654, −12.52662475985010, −11.81568395728113, −11.24077095394047, −10.84008604489331, −10.19056300930933, −9.725744831104435, −9.350002148441721, −8.349506735070373, −8.016765443786867, −7.180288201634932, −6.563831701811754, −6.208880521261879, −5.815544133191898, −5.112730223766233, −4.454331822415490, −3.996868422497428, −3.166455333310515, −2.650340219193025, −1.818272605736305, −1.041012762338478, 0,
1.041012762338478, 1.818272605736305, 2.650340219193025, 3.166455333310515, 3.996868422497428, 4.454331822415490, 5.112730223766233, 5.815544133191898, 6.208880521261879, 6.563831701811754, 7.180288201634932, 8.016765443786867, 8.349506735070373, 9.350002148441721, 9.725744831104435, 10.19056300930933, 10.84008604489331, 11.24077095394047, 11.81568395728113, 12.52662475985010, 12.72063932410654, 13.18924047285203, 13.80726118952655, 14.26863342255615, 14.78705540282558
