| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·11-s + 13-s − 2·14-s + 16-s − 4·17-s − 3·19-s + 2·22-s − 6·23-s + 26-s − 2·28-s + 29-s + 32-s − 4·34-s − 37-s − 3·38-s + 5·41-s + 4·43-s + 2·44-s − 6·46-s − 3·47-s − 3·49-s + 52-s − 7·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.603·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.688·19-s + 0.426·22-s − 1.25·23-s + 0.196·26-s − 0.377·28-s + 0.185·29-s + 0.176·32-s − 0.685·34-s − 0.164·37-s − 0.486·38-s + 0.780·41-s + 0.609·43-s + 0.301·44-s − 0.884·46-s − 0.437·47-s − 3/7·49-s + 0.138·52-s − 0.961·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.62827939484257358880497976769, −6.78611864725740928394076862537, −6.25939187918014826036710352588, −5.79664150305420381718530019255, −4.61946920885709173117332402900, −4.14401725933845394886243888442, −3.34221576889949835479739133545, −2.47916836811530617685360218334, −1.54785205526091033214844448364, 0,
1.54785205526091033214844448364, 2.47916836811530617685360218334, 3.34221576889949835479739133545, 4.14401725933845394886243888442, 4.61946920885709173117332402900, 5.79664150305420381718530019255, 6.25939187918014826036710352588, 6.78611864725740928394076862537, 7.62827939484257358880497976769
