| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 7-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 6·13-s + 14-s + 15-s − 16-s + 17-s − 18-s − 4·19-s − 20-s − 21-s − 4·22-s + 3·24-s + 25-s − 6·26-s + 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.547848366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547848366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.259792231759718199406518934146, −8.554941065667881198152640132576, −8.148987566119781843092589530959, −6.87580607524824807582513630110, −6.34452255010451568241792947625, −5.21436961317717372945251887956, −4.01753637196409798628019358183, −3.53735286344356544160995058129, −1.95408258983633536304488564831, −1.00374180151249186680119240710,
1.00374180151249186680119240710, 1.95408258983633536304488564831, 3.53735286344356544160995058129, 4.01753637196409798628019358183, 5.21436961317717372945251887956, 6.34452255010451568241792947625, 6.87580607524824807582513630110, 8.148987566119781843092589530959, 8.554941065667881198152640132576, 9.259792231759718199406518934146
