L(s) = 1 | − 2-s + 4-s − 4·5-s − 7-s − 8-s + 4·10-s − 4·11-s + 14-s + 16-s + 17-s + 2·19-s − 4·20-s + 4·22-s + 7·23-s + 11·25-s − 28-s − 10·29-s − 5·31-s − 32-s − 34-s + 4·35-s + 2·37-s − 2·38-s + 4·40-s + 2·41-s + 6·43-s − 4·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s + 1.26·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.894·20-s + 0.852·22-s + 1.45·23-s + 11/5·25-s − 0.188·28-s − 1.85·29-s − 0.898·31-s − 0.176·32-s − 0.171·34-s + 0.676·35-s + 0.328·37-s − 0.324·38-s + 0.632·40-s + 0.312·41-s + 0.914·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−14.79606563883795, −14.64859274177498, −13.58766389356339, −12.99113824888481, −12.59961922537141, −12.25085113680358, −11.33272088947130, −11.14228495728592, −10.87541463172556, −10.08690391004624, −9.432081757724673, −9.020345529711999, −8.395054228639809, −7.712243729424277, −7.610059405715630, −7.100596696551197, −6.452136993299975, −5.470837164228661, −5.190711297171537, −4.275821108777570, −3.722707926279229, −3.073720965539513, −2.670763151328659, −1.559763760750261, −0.6299649649662659, 0,
0.6299649649662659, 1.559763760750261, 2.670763151328659, 3.073720965539513, 3.722707926279229, 4.275821108777570, 5.190711297171537, 5.470837164228661, 6.452136993299975, 7.100596696551197, 7.610059405715630, 7.712243729424277, 8.395054228639809, 9.020345529711999, 9.432081757724673, 10.08690391004624, 10.87541463172556, 11.14228495728592, 11.33272088947130, 12.25085113680358, 12.59961922537141, 12.99113824888481, 13.58766389356339, 14.64859274177498, 14.79606563883795