L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 3·14-s − 15-s + 16-s + 6·17-s − 18-s + 3·19-s − 20-s − 3·21-s + 4·23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + 2·29-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 − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s − 0.654·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + 0.371·29-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 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.12904030534786, −14.29017308351667, −13.83736871112252, −13.38414235585725, −12.65839343060265, −12.21218007245451, −11.90534759318903, −11.12011255967144, −10.51524235836190, −10.04486299440972, −9.616258128956582, −9.152332290551537, −8.477781512867913, −8.115830307979058, −7.464626309061020, −6.968293068578033, −6.515596064112379, −5.777993533909052, −5.164517100975139, −4.341766776398973, −3.480615404736271, −3.155610614251765, −2.718959503334829, −1.552533843690782, −0.9838998185931907, 0,
0.9838998185931907, 1.552533843690782, 2.718959503334829, 3.155610614251765, 3.480615404736271, 4.341766776398973, 5.164517100975139, 5.777993533909052, 6.515596064112379, 6.968293068578033, 7.464626309061020, 8.115830307979058, 8.477781512867913, 9.152332290551537, 9.616258128956582, 10.04486299440972, 10.51524235836190, 11.12011255967144, 11.90534759318903, 12.21218007245451, 12.65839343060265, 13.38414235585725, 13.83736871112252, 14.29017308351667, 15.12904030534786