| L(s) = 1 | + 3-s − 2·5-s + 2·7-s + 9-s + 11-s + 2·13-s − 2·15-s + 4·17-s − 6·19-s + 2·21-s − 25-s + 27-s + 8·29-s + 8·31-s + 33-s − 4·35-s − 10·37-s + 2·39-s + 8·41-s − 2·43-s − 2·45-s + 8·47-s − 3·49-s + 4·51-s + 2·53-s − 2·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.174·33-s − 0.676·35-s − 1.64·37-s + 0.320·39-s + 1.24·41-s − 0.304·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.149774055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149774055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.737218941084006795512675250399, −8.363076200036620814816094088030, −7.77111220608894662838123356618, −6.88999788392414596472432742699, −6.02794795488925464471388657825, −4.84093273741330942527412777536, −4.14647488588500988181459781943, −3.38179971239828744260506351289, −2.23512262983214968168477739558, −0.989132441220879554426759426737,
0.989132441220879554426759426737, 2.23512262983214968168477739558, 3.38179971239828744260506351289, 4.14647488588500988181459781943, 4.84093273741330942527412777536, 6.02794795488925464471388657825, 6.88999788392414596472432742699, 7.77111220608894662838123356618, 8.363076200036620814816094088030, 8.737218941084006795512675250399
