| L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s + 2·13-s + 6·17-s + 21-s − 8·23-s − 27-s + 10·29-s + 8·31-s − 4·33-s − 2·37-s − 2·39-s − 2·41-s + 8·43-s + 4·47-s + 49-s − 6·51-s − 10·53-s − 4·59-s − 6·61-s − 63-s + 8·69-s + 12·71-s + 6·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.218·21-s − 1.66·23-s − 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.520·59-s − 0.768·61-s − 0.125·63-s + 0.963·69-s + 1.42·71-s + 0.702·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.942655813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942655813\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.943552742381186130031454597567, −6.88880949229450837493696086147, −6.29583658350568991035172706042, −5.97375595059385509201334142005, −5.02578405727115806900791402051, −4.21046512009068981338244284230, −3.62697049484495173523243891598, −2.72922325965124720845675524074, −1.49770912323389752038351184824, −0.77189749372046361869451260745,
0.77189749372046361869451260745, 1.49770912323389752038351184824, 2.72922325965124720845675524074, 3.62697049484495173523243891598, 4.21046512009068981338244284230, 5.02578405727115806900791402051, 5.97375595059385509201334142005, 6.29583658350568991035172706042, 6.88880949229450837493696086147, 7.943552742381186130031454597567
