L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s − 17-s + 5·19-s − 20-s − 5·23-s − 4·25-s + 2·26-s + 28-s − 2·29-s − 31-s + 32-s − 34-s − 35-s + 37-s + 5·38-s − 40-s + 6·41-s − 12·43-s − 5·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.14·19-s − 0.223·20-s − 1.04·23-s − 4/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s + 0.164·37-s + 0.811·38-s − 0.158·40-s + 0.937·41-s − 1.82·43-s − 0.737·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.92894208069401, −12.79936644921481, −11.88387178891786, −11.73761389920870, −11.44369050298382, −10.93174804156257, −10.26115505119941, −9.949816856806844, −9.398787253145600, −8.778111585409567, −8.187787464561668, −7.973566175466936, −7.253384129321391, −7.062308160495767, −6.226342157283219, −5.923855884577349, −5.344681113576548, −4.928364948075899, −4.223783677598240, −3.896105704207616, −3.392009912390269, −2.822262857757904, −2.072116130754741, −1.620355833011532, −0.8728409970989701, 0,
0.8728409970989701, 1.620355833011532, 2.072116130754741, 2.822262857757904, 3.392009912390269, 3.896105704207616, 4.223783677598240, 4.928364948075899, 5.344681113576548, 5.923855884577349, 6.226342157283219, 7.062308160495767, 7.253384129321391, 7.973566175466936, 8.187787464561668, 8.778111585409567, 9.398787253145600, 9.949816856806844, 10.26115505119941, 10.93174804156257, 11.44369050298382, 11.73761389920870, 11.88387178891786, 12.79936644921481, 12.92894208069401