| L(s) = 1 | − 3-s − 5-s + 9-s − 5·11-s + 13-s + 15-s + 17-s − 6·19-s − 6·23-s − 4·25-s − 27-s + 6·29-s + 4·31-s + 5·33-s + 11·37-s − 39-s + 9·43-s − 45-s + 4·47-s − 51-s − 7·53-s + 5·55-s + 6·57-s + 12·59-s − 6·61-s − 65-s − 13·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.870·33-s + 1.80·37-s − 0.160·39-s + 1.37·43-s − 0.149·45-s + 0.583·47-s − 0.140·51-s − 0.961·53-s + 0.674·55-s + 0.794·57-s + 1.56·59-s − 0.768·61-s − 0.124·65-s − 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6565446994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6565446994\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.88330459391145, −14.28102012764488, −13.60417680662097, −13.23688684040866, −12.56144835778578, −12.28914340850022, −11.62049737268715, −11.12113888179397, −10.55152251122202, −10.21495504604008, −9.673453790757077, −8.870826369799453, −8.133161756122917, −7.922175175509099, −7.385421327420824, −6.470259155247249, −6.071309795873701, −5.590651409961107, −4.760909204460497, −4.300969669216448, −3.780357491605326, −2.660977091464302, −2.392953513705318, −1.272252162044196, −0.3201293506361209,
0.3201293506361209, 1.272252162044196, 2.392953513705318, 2.660977091464302, 3.780357491605326, 4.300969669216448, 4.760909204460497, 5.590651409961107, 6.071309795873701, 6.470259155247249, 7.385421327420824, 7.922175175509099, 8.133161756122917, 8.870826369799453, 9.673453790757077, 10.21495504604008, 10.55152251122202, 11.12113888179397, 11.62049737268715, 12.28914340850022, 12.56144835778578, 13.23688684040866, 13.60417680662097, 14.28102012764488, 14.88330459391145
