| L(s) = 1 | − 3-s + 9-s − 2·13-s + 6·17-s − 4·19-s − 4·23-s − 27-s + 6·29-s − 8·31-s + 10·37-s + 2·39-s + 10·41-s + 12·43-s + 8·47-s − 6·51-s − 6·53-s + 4·57-s + 4·59-s + 10·61-s + 12·67-s + 4·69-s − 4·71-s + 2·73-s − 8·79-s + 81-s − 4·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s + 1.16·47-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s + 1.28·61-s + 1.46·67-s + 0.481·69-s − 0.474·71-s + 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962464907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962464907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.27639978066971, −14.14974345750449, −13.05866381022046, −12.70833272209965, −12.42149641873816, −11.84469929243545, −11.21343448012464, −10.86020493019540, −10.22848844305156, −9.764555153969442, −9.365277847478310, −8.590663186715346, −8.048373854002216, −7.406530713237354, −7.187120265853914, −6.151682458786084, −5.956573485785769, −5.401311555172147, −4.628597684586532, −4.150042184384228, −3.578500559850136, −2.604682447876504, −2.205427527105163, −1.137401801715649, −0.5653563546612011,
0.5653563546612011, 1.137401801715649, 2.205427527105163, 2.604682447876504, 3.578500559850136, 4.150042184384228, 4.628597684586532, 5.401311555172147, 5.956573485785769, 6.151682458786084, 7.187120265853914, 7.406530713237354, 8.048373854002216, 8.590663186715346, 9.365277847478310, 9.764555153969442, 10.22848844305156, 10.86020493019540, 11.21343448012464, 11.84469929243545, 12.42149641873816, 12.70833272209965, 13.05866381022046, 14.14974345750449, 14.27639978066971
