| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s + 11-s + 2·13-s − 2·15-s + 4·17-s + 2·21-s − 25-s − 27-s + 8·29-s + 8·31-s − 33-s − 4·35-s − 10·37-s − 2·39-s − 8·41-s − 2·43-s + 2·45-s − 8·47-s − 3·49-s − 4·51-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s + 0.970·17-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47652 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47652 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.092436414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092436414\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.49928018634908, −13.87273452743896, −13.55950583299596, −13.18364966344240, −12.39684789501320, −11.96476926198673, −11.74801813523446, −10.74022832053055, −10.39179528186433, −9.915485765076417, −9.581879688157541, −8.842540270089539, −8.306536416806628, −7.710692200926793, −6.784823308949462, −6.488843557735878, −6.125159846443922, −5.392462868265561, −4.953158326219969, −4.221243989706868, −3.296769444483808, −3.071254208839258, −1.947305483749334, −1.391343434902428, −0.5508987710360389,
0.5508987710360389, 1.391343434902428, 1.947305483749334, 3.071254208839258, 3.296769444483808, 4.221243989706868, 4.953158326219969, 5.392462868265561, 6.125159846443922, 6.488843557735878, 6.784823308949462, 7.710692200926793, 8.306536416806628, 8.842540270089539, 9.581879688157541, 9.915485765076417, 10.39179528186433, 10.74022832053055, 11.74801813523446, 11.96476926198673, 12.39684789501320, 13.18364966344240, 13.55950583299596, 13.87273452743896, 14.49928018634908
