L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s − 14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s − 20-s − 21-s + 24-s + 25-s − 2·26-s − 27-s + 28-s − 30-s + 4·31-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 + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.478450470\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478450470\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.18721016025508, −12.34486898088749, −12.11037517227634, −11.81612649078403, −11.28353798160740, −10.79314651314228, −10.27875004481591, −10.04751455817779, −9.336671496771968, −8.934172083549907, −8.295735369819998, −7.854430802647489, −7.562665318512448, −6.897838219524909, −6.509584185848202, −5.793534898160269, −5.406222370625980, −4.927091996063599, −4.203184419708981, −3.560304825612146, −3.155323039832001, −2.404944661447174, −1.494905521471232, −1.150194803546410, −0.4662814590554064,
0.4662814590554064, 1.150194803546410, 1.494905521471232, 2.404944661447174, 3.155323039832001, 3.560304825612146, 4.203184419708981, 4.927091996063599, 5.406222370625980, 5.793534898160269, 6.509584185848202, 6.897838219524909, 7.562665318512448, 7.854430802647489, 8.295735369819998, 8.934172083549907, 9.336671496771968, 10.04751455817779, 10.27875004481591, 10.79314651314228, 11.28353798160740, 11.81612649078403, 12.11037517227634, 12.34486898088749, 13.18721016025508