| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s − 2·13-s + 15-s + 2·17-s + 4·19-s − 21-s − 8·23-s + 25-s + 27-s + 6·29-s − 8·31-s + 33-s − 35-s − 2·37-s − 2·39-s + 2·41-s − 4·43-s + 45-s − 8·47-s + 49-s + 2·51-s − 10·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−16.10835370655519, −15.59145987170723, −14.68508813414481, −14.39912244562548, −13.99175097277544, −13.33989237010713, −12.79892136909136, −12.21912307645434, −11.74905569410654, −11.04200628460582, −10.11291683641455, −9.933911334008517, −9.426003712066396, −8.710305527947371, −8.124101736155727, −7.512871721196083, −6.915491062357059, −6.257643104789036, −5.592020302701296, −4.959474039595674, −4.127877581876984, −3.447785014992984, −2.834904176245048, −2.001310813726218, −1.297360848074301, 0,
1.297360848074301, 2.001310813726218, 2.834904176245048, 3.447785014992984, 4.127877581876984, 4.959474039595674, 5.592020302701296, 6.257643104789036, 6.915491062357059, 7.512871721196083, 8.124101736155727, 8.710305527947371, 9.426003712066396, 9.933911334008517, 10.11291683641455, 11.04200628460582, 11.74905569410654, 12.21912307645434, 12.79892136909136, 13.33989237010713, 13.99175097277544, 14.39912244562548, 14.68508813414481, 15.59145987170723, 16.10835370655519
