L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s + 2·13-s − 4·14-s + 16-s − 2·17-s + 19-s − 20-s + 8·23-s + 25-s + 2·26-s − 4·28-s + 6·29-s + 4·31-s + 32-s − 2·34-s + 4·35-s − 10·37-s + 38-s − 40-s − 2·41-s − 12·43-s + 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.676·35-s − 1.64·37-s + 0.162·38-s − 0.158·40-s − 0.312·41-s − 1.82·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.21137155596650, −12.93124764450892, −12.36481788531927, −11.95394961080796, −11.54805581111794, −10.92132371442916, −10.56833868305468, −9.983821453128020, −9.651669523451773, −8.871241311663621, −8.640876124471614, −8.058440544063164, −7.308238610937515, −6.825257656640982, −6.549062847904208, −6.235335296104934, −5.307851862468701, −5.065650703890187, −4.434007308299283, −3.741084008620057, −3.317243022015215, −3.022916105197944, −2.381529959808171, −1.497481339173784, −0.7984099183289185, 0,
0.7984099183289185, 1.497481339173784, 2.381529959808171, 3.022916105197944, 3.317243022015215, 3.741084008620057, 4.434007308299283, 5.065650703890187, 5.307851862468701, 6.235335296104934, 6.549062847904208, 6.825257656640982, 7.308238610937515, 8.058440544063164, 8.640876124471614, 8.871241311663621, 9.651669523451773, 9.983821453128020, 10.56833868305468, 10.92132371442916, 11.54805581111794, 11.95394961080796, 12.36481788531927, 12.93124764450892, 13.21137155596650