| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 2·13-s + 6·19-s + 21-s + 23-s − 27-s + 29-s − 2·31-s + 33-s − 7·37-s + 2·39-s − 8·41-s − 43-s − 2·47-s + 49-s − 14·53-s − 6·57-s + 10·59-s − 63-s − 3·67-s − 69-s − 9·71-s + 77-s + 79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.37·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.185·29-s − 0.359·31-s + 0.174·33-s − 1.15·37-s + 0.320·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s − 1.92·53-s − 0.794·57-s + 1.30·59-s − 0.125·63-s − 0.366·67-s − 0.120·69-s − 1.06·71-s + 0.113·77-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.781529339256990613587869029567, −7.80298801990183034032490335907, −7.10913822752986093173591589044, −6.40155300612741441209080116015, −5.37994560739225437445244358999, −4.93189704452359005958027193915, −3.71059370521019590152341252906, −2.82842343595315456135468530958, −1.47588350858440766249674578424, 0,
1.47588350858440766249674578424, 2.82842343595315456135468530958, 3.71059370521019590152341252906, 4.93189704452359005958027193915, 5.37994560739225437445244358999, 6.40155300612741441209080116015, 7.10913822752986093173591589044, 7.80298801990183034032490335907, 8.781529339256990613587869029567
