| L(s) = 1 | − 2·5-s − 4·11-s + 2·13-s − 17-s − 4·19-s − 25-s − 10·29-s + 8·31-s + 2·37-s − 10·41-s − 12·43-s − 7·49-s + 6·53-s + 8·55-s + 12·59-s + 10·61-s − 4·65-s + 12·67-s + 10·73-s − 8·79-s + 4·83-s + 2·85-s + 6·89-s + 8·95-s − 14·97-s − 10·101-s − 8·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.328·37-s − 1.56·41-s − 1.82·43-s − 49-s + 0.824·53-s + 1.07·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s + 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s + 0.820·95-s − 1.42·97-s − 0.995·101-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8111745406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8111745406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.86181043713461154412505593216, −6.94773718046313634845319465223, −6.48207232670741268996984357376, −5.46584965257912850606053369991, −4.99880483036204675913788063859, −4.02567641236163563211535902605, −3.58943662862194459410078243045, −2.61414311348008518376390369912, −1.79115703054004385997644970791, −0.41134430024497760893483927350,
0.41134430024497760893483927350, 1.79115703054004385997644970791, 2.61414311348008518376390369912, 3.58943662862194459410078243045, 4.02567641236163563211535902605, 4.99880483036204675913788063859, 5.46584965257912850606053369991, 6.48207232670741268996984357376, 6.94773718046313634845319465223, 7.86181043713461154412505593216
