L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s + 2·13-s − 15-s − 2·17-s − 21-s + 4·23-s + 25-s + 27-s − 2·29-s + 4·31-s + 4·33-s + 35-s − 10·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 6·53-s − 4·55-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 0.169·35-s − 1.64·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.539·55-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066057259\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066057259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.078478534128429145164100410087, −8.791237224551889431601784810184, −7.80780000994835223632519133816, −6.91987606937710160284722602604, −6.37952088717010441136974956603, −5.18052200208689310723232042498, −4.04867971009360347613759559617, −3.55267890775103657730631859597, −2.37390768061023031834660486712, −1.02197690021927726048891601742,
1.02197690021927726048891601742, 2.37390768061023031834660486712, 3.55267890775103657730631859597, 4.04867971009360347613759559617, 5.18052200208689310723232042498, 6.37952088717010441136974956603, 6.91987606937710160284722602604, 7.80780000994835223632519133816, 8.791237224551889431601784810184, 9.078478534128429145164100410087