| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s − 2·9-s + 10-s + 2·11-s − 12-s − 2·13-s − 14-s − 15-s + 16-s − 17-s − 2·18-s + 7·19-s + 20-s + 21-s + 2·22-s − 4·23-s − 24-s − 4·25-s − 2·26-s + 5·27-s − 28-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 + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.392·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.00306424191082, −13.33457010378476, −12.87786027399982, −12.28903115202011, −11.85904134422632, −11.48979574094221, −11.27155312831942, −10.24321858881622, −10.12679496188135, −9.481618519171137, −9.126474066630423, −8.218058718617120, −7.882738594216463, −7.256663181496927, −6.496511931823459, −6.269666417455736, −5.846662514297031, −5.242868283521742, −4.709567707013666, −4.279191226649191, −3.408130274706056, −2.967961273512136, −2.445529448939040, −1.602092228686851, −0.9208844135737929, 0,
0.9208844135737929, 1.602092228686851, 2.445529448939040, 2.967961273512136, 3.408130274706056, 4.279191226649191, 4.709567707013666, 5.242868283521742, 5.846662514297031, 6.269666417455736, 6.496511931823459, 7.256663181496927, 7.882738594216463, 8.218058718617120, 9.126474066630423, 9.481618519171137, 10.12679496188135, 10.24321858881622, 11.27155312831942, 11.48979574094221, 11.85904134422632, 12.28903115202011, 12.87786027399982, 13.33457010378476, 14.00306424191082
