| L(s) = 1 | − 3-s − 3·5-s − 2·7-s + 9-s − 2·11-s + 3·15-s − 3·17-s + 6·19-s + 2·21-s + 6·23-s + 4·25-s − 27-s + 5·29-s + 2·33-s + 6·35-s − 37-s + 5·41-s − 6·43-s − 3·45-s + 6·47-s − 3·49-s + 3·51-s − 3·53-s + 6·55-s − 6·57-s − 12·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.774·15-s − 0.727·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.928·29-s + 0.348·33-s + 1.01·35-s − 0.164·37-s + 0.780·41-s − 0.914·43-s − 0.447·45-s + 0.875·47-s − 3/7·49-s + 0.420·51-s − 0.412·53-s + 0.809·55-s − 0.794·57-s − 1.56·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7950856343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7950856343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.18608576270903, −14.73903324699246, −13.79146149739529, −13.45238221861696, −12.81193512087381, −12.29207414551192, −11.92663427078657, −11.29179588511962, −10.92514432323083, −10.35302051235792, −9.697202409533379, −9.103232644903556, −8.560684382313789, −7.688677288248735, −7.524339982667742, −6.791147485654931, −6.318258815157785, −5.537603159366003, −4.788040482961421, −4.532020010699420, −3.442149293080033, −3.269011575758793, −2.367651849681802, −1.119309690738547, −0.4067088130758893,
0.4067088130758893, 1.119309690738547, 2.367651849681802, 3.269011575758793, 3.442149293080033, 4.532020010699420, 4.788040482961421, 5.537603159366003, 6.318258815157785, 6.791147485654931, 7.524339982667742, 7.688677288248735, 8.560684382313789, 9.103232644903556, 9.697202409533379, 10.35302051235792, 10.92514432323083, 11.29179588511962, 11.92663427078657, 12.29207414551192, 12.81193512087381, 13.45238221861696, 13.79146149739529, 14.73903324699246, 15.18608576270903
