| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s + 4·14-s − 15-s + 16-s − 17-s + 18-s + 8·19-s + 20-s − 4·21-s − 4·22-s − 4·23-s − 24-s + 25-s + 26-s − 27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.600144062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.600144062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.86824490073642232309911337797, −7.32385195395029597181277860627, −6.39795257329275366013853739358, −5.60677637359400498054372352493, −5.16569518078440456499364753179, −4.69862484714876694406047156042, −3.74230288344547264208452304737, −2.68538743597802023975597753772, −1.89005371980077097204425647474, −0.949712077093368449409452845915,
0.949712077093368449409452845915, 1.89005371980077097204425647474, 2.68538743597802023975597753772, 3.74230288344547264208452304737, 4.69862484714876694406047156042, 5.16569518078440456499364753179, 5.60677637359400498054372352493, 6.39795257329275366013853739358, 7.32385195395029597181277860627, 7.86824490073642232309911337797
