L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 4·11-s + 13-s − 4·14-s + 16-s − 5·17-s + 20-s − 4·22-s + 3·23-s + 25-s + 26-s − 4·28-s + 2·29-s + 31-s + 32-s − 5·34-s − 4·35-s + 7·37-s + 40-s − 8·41-s − 4·44-s + 3·46-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 − 1.20·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 1.21·17-s + 0.223·20-s − 0.852·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s + 0.371·29-s + 0.179·31-s + 0.176·32-s − 0.857·34-s − 0.676·35-s + 1.15·37-s + 0.158·40-s − 1.24·41-s − 0.603·44-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.43220639445423, −13.11295318060993, −12.65266245817763, −12.33574923434660, −11.54521994606401, −11.17670030156455, −10.57146547065331, −10.20138682193548, −9.815776965945636, −9.167485522868790, −8.786351532518350, −8.148413315167165, −7.549812135410981, −6.955692108649730, −6.496804606800057, −6.258379908014875, −5.541574071344879, −5.167797570674022, −4.528247820854466, −3.974901001681810, −3.314480977060958, −2.775011326586702, −2.500995087013698, −1.729813222861155, −0.7612430977250567, 0,
0.7612430977250567, 1.729813222861155, 2.500995087013698, 2.775011326586702, 3.314480977060958, 3.974901001681810, 4.528247820854466, 5.167797570674022, 5.541574071344879, 6.258379908014875, 6.496804606800057, 6.955692108649730, 7.549812135410981, 8.148413315167165, 8.786351532518350, 9.167485522868790, 9.815776965945636, 10.20138682193548, 10.57146547065331, 11.17670030156455, 11.54521994606401, 12.33574923434660, 12.65266245817763, 13.11295318060993, 13.43220639445423