| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s + 2·11-s − 13-s − 2·15-s + 6·17-s − 2·19-s + 2·21-s − 25-s − 27-s + 2·29-s + 2·31-s − 2·33-s − 4·35-s + 10·37-s + 39-s + 2·41-s + 8·43-s + 2·45-s − 2·47-s − 3·49-s − 6·51-s − 2·53-s + 4·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.676·35-s + 1.64·37-s + 0.160·39-s + 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.291·47-s − 3/7·49-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.541529812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541529812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.842514528404132269540859207536, −9.168140770096584540702033093028, −8.016052793793976190414618806262, −7.06541986235410893916933421520, −6.14841639240660146897531624692, −5.78517701336628451955041513916, −4.65680782039109928862677817630, −3.55810218913298118185059814514, −2.36636166197971073120579022543, −0.983919193523929230329764658751,
0.983919193523929230329764658751, 2.36636166197971073120579022543, 3.55810218913298118185059814514, 4.65680782039109928862677817630, 5.78517701336628451955041513916, 6.14841639240660146897531624692, 7.06541986235410893916933421520, 8.016052793793976190414618806262, 9.168140770096584540702033093028, 9.842514528404132269540859207536
