L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 12-s + 6·13-s − 15-s − 16-s − 18-s − 4·19-s + 20-s + 3·24-s + 25-s − 6·26-s + 27-s + 2·29-s + 30-s − 5·32-s − 36-s + 6·37-s + 4·38-s + 6·39-s − 3·40-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.883·32-s − 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.960·39-s − 0.474·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.25384151436251, −12.97063775216524, −12.33116820842411, −11.78988158353711, −11.21883755363805, −10.80795076768606, −10.27230985021937, −10.08521282554944, −9.194377936600174, −8.974086159997587, −8.530135316041587, −8.251250858367649, −7.616079005777383, −7.346816619982047, −6.511627479027732, −6.180092878098436, −5.533042438114129, −4.755043251868245, −4.291991808546731, −3.998750635695289, −3.255546484133713, −2.882882534885343, −1.828193571753863, −1.488780999470562, −0.7711296133260378, 0,
0.7711296133260378, 1.488780999470562, 1.828193571753863, 2.882882534885343, 3.255546484133713, 3.998750635695289, 4.291991808546731, 4.755043251868245, 5.533042438114129, 6.180092878098436, 6.511627479027732, 7.346816619982047, 7.616079005777383, 8.251250858367649, 8.530135316041587, 8.974086159997587, 9.194377936600174, 10.08521282554944, 10.27230985021937, 10.80795076768606, 11.21883755363805, 11.78988158353711, 12.33116820842411, 12.97063775216524, 13.25384151436251