L(s) = 1 | − 2·4-s − 3·5-s + 7-s − 11-s + 4·13-s + 4·16-s + 6·20-s − 6·23-s + 4·25-s − 2·28-s + 6·29-s − 5·31-s − 3·35-s − 8·37-s − 9·41-s + 2·43-s + 2·44-s − 9·47-s + 49-s − 8·52-s − 3·53-s + 3·55-s − 7·61-s − 8·64-s − 12·65-s + 4·67-s − 9·71-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s + 16-s + 1.34·20-s − 1.25·23-s + 4/5·25-s − 0.377·28-s + 1.11·29-s − 0.898·31-s − 0.507·35-s − 1.31·37-s − 1.40·41-s + 0.304·43-s + 0.301·44-s − 1.31·47-s + 1/7·49-s − 1.10·52-s − 0.412·53-s + 0.404·55-s − 0.896·61-s − 64-s − 1.48·65-s + 0.488·67-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.14392513615330, −12.49627704045474, −12.22099121098112, −11.69973189773500, −11.34241065634121, −10.76220481393329, −10.27430028848973, −9.994110990033475, −9.152517176051775, −8.812423476726031, −8.325922676130418, −8.002816509422715, −7.732252557611551, −6.944400147714306, −6.532163379737751, −5.769495793188626, −5.387704742492362, −4.744052936730951, −4.322797298200779, −3.917158200692196, −3.318751771419579, −3.085855881498889, −1.879325010789506, −1.441237369104189, −0.5483932096233678, 0,
0.5483932096233678, 1.441237369104189, 1.879325010789506, 3.085855881498889, 3.318751771419579, 3.917158200692196, 4.322797298200779, 4.744052936730951, 5.387704742492362, 5.769495793188626, 6.532163379737751, 6.944400147714306, 7.732252557611551, 8.002816509422715, 8.325922676130418, 8.812423476726031, 9.152517176051775, 9.994110990033475, 10.27430028848973, 10.76220481393329, 11.34241065634121, 11.69973189773500, 12.22099121098112, 12.49627704045474, 13.14392513615330