| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 6·11-s + 2·13-s + 15-s − 2·17-s + 2·19-s − 2·21-s + 6·23-s + 25-s + 27-s − 10·29-s − 4·31-s + 6·33-s − 2·35-s − 2·37-s + 2·39-s + 6·41-s + 45-s + 6·47-s − 3·49-s − 2·51-s + 10·53-s + 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 1.04·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s + 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.498641663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498641663\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.333027182762793086552952191466, −8.740725234883061553130030026329, −7.53186231287917567143611986444, −6.79551474820332812646473796605, −6.20240643140516046944179734607, −5.21420829100854825114259578502, −3.91562711770929450018042729419, −3.49090121852027045476332300058, −2.22910019639663757261053674603, −1.12397443494594220839491514104,
1.12397443494594220839491514104, 2.22910019639663757261053674603, 3.49090121852027045476332300058, 3.91562711770929450018042729419, 5.21420829100854825114259578502, 6.20240643140516046944179734607, 6.79551474820332812646473796605, 7.53186231287917567143611986444, 8.740725234883061553130030026329, 9.333027182762793086552952191466
