| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 14-s − 2·15-s + 16-s + 6·17-s + 18-s − 2·20-s − 21-s + 4·22-s + 23-s + 24-s − 25-s + 2·26-s + 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.785859174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785859174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−19.59898598328709, −19.15243637764348, −18.45713555924502, −17.28482013004122, −16.48532021785977, −15.99411594496336, −15.12325736702405, −14.64921142003217, −13.92136289670841, −13.19544569932966, −12.27803632902950, −11.83355038125173, −11.02155838848655, −9.968594132488272, −9.190747247293543, −8.181256656550651, −7.520501337946085, −6.594287683128709, −5.733340524802486, −4.393767706045575, −3.721751174960615, −2.977987879245730, −1.317545632041997,
1.317545632041997, 2.977987879245730, 3.721751174960615, 4.393767706045575, 5.733340524802486, 6.594287683128709, 7.520501337946085, 8.181256656550651, 9.190747247293543, 9.968594132488272, 11.02155838848655, 11.83355038125173, 12.27803632902950, 13.19544569932966, 13.92136289670841, 14.64921142003217, 15.12325736702405, 15.99411594496336, 16.48532021785977, 17.28482013004122, 18.45713555924502, 19.15243637764348, 19.59898598328709
