L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 25-s − 27-s − 2·29-s + 4·31-s − 4·33-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s + 2·45-s + 8·47-s − 6·51-s − 10·53-s + 8·55-s − 4·57-s + 4·59-s + 6·61-s − 4·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 0.840·51-s − 1.37·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.504601491\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.504601491\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
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 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.58369824069551776391804250294, −6.92428553414297385286322411981, −6.26857371316841959262674737462, −5.61122489506948776314215410858, −5.20290884372379211655695572476, −4.23042694073593217824157523890, −3.50061205349313469932624989205, −2.54243246208562779714008279611, −1.54969288210031137888596853471, −0.853215550297782314362756939852,
0.853215550297782314362756939852, 1.54969288210031137888596853471, 2.54243246208562779714008279611, 3.50061205349313469932624989205, 4.23042694073593217824157523890, 5.20290884372379211655695572476, 5.61122489506948776314215410858, 6.26857371316841959262674737462, 6.92428553414297385286322411981, 7.58369824069551776391804250294