| L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s − 4·7-s + 8-s + 9-s − 10-s − 2·11-s + 2·12-s − 13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 8·21-s − 2·22-s + 6·23-s + 2·24-s + 25-s − 26-s − 4·27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 1.74·21-s − 0.426·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773094903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773094903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.26679558606257756188931391942, −12.71202338625229541571828648444, −11.47889106396016028805556897933, −10.04470252993954468916012522294, −9.161137327460865013114506016469, −7.83531743824800124245932651769, −6.87551315707654393130092819540, −5.33976892050247260934145252130, −3.53009310754873249134401547682, −2.88253971705917726242541361642,
2.88253971705917726242541361642, 3.53009310754873249134401547682, 5.33976892050247260934145252130, 6.87551315707654393130092819540, 7.83531743824800124245932651769, 9.161137327460865013114506016469, 10.04470252993954468916012522294, 11.47889106396016028805556897933, 12.71202338625229541571828648444, 13.26679558606257756188931391942
