L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s − 2·17-s − 19-s − 21-s − 4·23-s − 25-s − 27-s − 2·29-s + 2·35-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s + 2·51-s − 10·53-s + 57-s − 12·59-s − 10·61-s + 63-s + 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.229·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.132·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.71598464208158780765361859731, −6.73456560568447262229305324722, −6.22963690808262854308155816900, −5.62557596616354097945717516585, −4.89870089264221382961397004508, −4.15948364763880779739713392620, −3.20381309730681673396081067438, −2.02653962005260481268813451036, −1.48127017826735144571644996023, 0,
1.48127017826735144571644996023, 2.02653962005260481268813451036, 3.20381309730681673396081067438, 4.15948364763880779739713392620, 4.89870089264221382961397004508, 5.62557596616354097945717516585, 6.22963690808262854308155816900, 6.73456560568447262229305324722, 7.71598464208158780765361859731