| L(s) = 1 | + 2·3-s + 7-s + 9-s − 2·11-s − 4·13-s + 2·17-s − 4·19-s + 2·21-s + 23-s − 4·27-s + 2·29-s + 4·31-s − 4·33-s − 8·37-s − 8·39-s − 2·41-s − 10·43-s − 12·47-s + 49-s + 4·51-s − 4·53-s − 8·57-s + 10·59-s + 14·61-s + 63-s − 10·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.696·33-s − 1.31·37-s − 1.28·39-s − 0.312·41-s − 1.52·43-s − 1.75·47-s + 1/7·49-s + 0.560·51-s − 0.549·53-s − 1.05·57-s + 1.30·59-s + 1.79·61-s + 0.125·63-s − 1.22·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549418020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549418020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.94399772616711, −12.48202927299733, −11.85718748100665, −11.53897321089379, −10.99461724491061, −10.24487746376052, −10.02364730033037, −9.712095764516377, −8.913259417992834, −8.574438621646884, −8.193430329963226, −7.838956563325823, −7.290644101699359, −6.726898803872113, −6.371970429135908, −5.394743388970183, −5.161837284386058, −4.674320263891695, −3.952666558712740, −3.464466398780763, −2.893595771088781, −2.458912841084770, −1.937842681308539, −1.366906264687049, −0.2897337868542694,
0.2897337868542694, 1.366906264687049, 1.937842681308539, 2.458912841084770, 2.893595771088781, 3.464466398780763, 3.952666558712740, 4.674320263891695, 5.161837284386058, 5.394743388970183, 6.371970429135908, 6.726898803872113, 7.290644101699359, 7.838956563325823, 8.193430329963226, 8.574438621646884, 8.913259417992834, 9.712095764516377, 10.02364730033037, 10.24487746376052, 10.99461724491061, 11.53897321089379, 11.85718748100665, 12.48202927299733, 12.94399772616711
