L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 6·17-s + 4·19-s − 23-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s + 8·47-s − 6·51-s − 6·53-s − 8·55-s − 4·57-s + 4·59-s − 10·61-s + 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 0.840·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.769607870\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769607870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.66532062672228, −12.31239329275891, −12.05076281853130, −11.72846389607822, −11.28752896330903, −10.69752786244334, −10.14827009687788, −9.792642158991737, −9.274904320539725, −8.809736728586443, −8.034180621471110, −7.791079861040377, −7.273412547900058, −6.838091420316203, −6.240016585947941, −5.709076423198807, −5.320176677659584, −4.535314030579514, −4.244096031520385, −3.673393953343792, −3.113404606283938, −2.568762504285280, −1.508412906310101, −1.126157949994292, −0.4436669396692323,
0.4436669396692323, 1.126157949994292, 1.508412906310101, 2.568762504285280, 3.113404606283938, 3.673393953343792, 4.244096031520385, 4.535314030579514, 5.320176677659584, 5.709076423198807, 6.240016585947941, 6.838091420316203, 7.273412547900058, 7.791079861040377, 8.034180621471110, 8.809736728586443, 9.274904320539725, 9.792642158991737, 10.14827009687788, 10.69752786244334, 11.28752896330903, 11.72846389607822, 12.05076281853130, 12.31239329275891, 12.66532062672228