| L(s) = 1 | + 3.20·3-s + 7.26·9-s + 4.20·11-s − 0.204·13-s + 5.06·17-s − 1.06·19-s + 2.14·23-s + 13.6·27-s − 7.47·29-s + 8.47·31-s + 13.4·33-s − 10.6·37-s − 0.654·39-s − 10.5·41-s + 8.26·43-s − 3.26·47-s + 16.2·51-s + 5.67·53-s − 3.40·57-s − 1.20·59-s − 1.65·61-s − 12.4·67-s + 6.85·69-s − 0.591·71-s + 4·73-s + 6.54·79-s + 22.0·81-s + ⋯ |
| L(s) = 1 | + 1.85·3-s + 2.42·9-s + 1.26·11-s − 0.0566·13-s + 1.22·17-s − 0.244·19-s + 0.446·23-s + 2.63·27-s − 1.38·29-s + 1.52·31-s + 2.34·33-s − 1.74·37-s − 0.104·39-s − 1.64·41-s + 1.26·43-s − 0.476·47-s + 2.27·51-s + 0.779·53-s − 0.451·57-s − 0.156·59-s − 0.211·61-s − 1.51·67-s + 0.825·69-s − 0.0701·71-s + 0.468·73-s + 0.736·79-s + 2.44·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.694478953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.694478953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.20T + 3T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 0.204T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + 1.20T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.591T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 + 3.88T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + 1.33T + 97T^{2} \) |
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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
−8.413547220655776541693624783234, −7.61540011897655682824630463868, −7.08570468313913497481575990335, −6.32002051565248116204196020800, −5.18438286160024668125836091872, −4.21634239445265908818637266635, −3.57486851815063695312542703647, −3.02414583807924712768416431294, −1.96029960676183560770132618758, −1.23324685342691919993915079721,
1.23324685342691919993915079721, 1.96029960676183560770132618758, 3.02414583807924712768416431294, 3.57486851815063695312542703647, 4.21634239445265908818637266635, 5.18438286160024668125836091872, 6.32002051565248116204196020800, 7.08570468313913497481575990335, 7.61540011897655682824630463868, 8.413547220655776541693624783234
