| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 11-s + 2·13-s + 2·15-s + 2·17-s − 4·19-s − 4·21-s − 25-s − 27-s + 6·29-s + 4·31-s − 33-s − 8·35-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s − 2·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.174·33-s − 1.35·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s + 0.529·57-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
| \(L(1)\) |
\(\approx\) |
\(1.500922204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500922204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.854053752616410884595967992719, −8.188005636037401794163293024214, −7.74008675811713615622276560519, −6.74194839458912705809412936401, −5.94818780103341542269279564846, −4.88724985058474154856715375096, −4.41246815642915893301829113030, −3.47762902586684244536957801064, −1.98432210663483889508619789323, −0.870188403122308180302100548115,
0.870188403122308180302100548115, 1.98432210663483889508619789323, 3.47762902586684244536957801064, 4.41246815642915893301829113030, 4.88724985058474154856715375096, 5.94818780103341542269279564846, 6.74194839458912705809412936401, 7.74008675811713615622276560519, 8.188005636037401794163293024214, 8.854053752616410884595967992719
