L(s) = 1 | + 2-s + 4-s + 8-s + 13-s + 16-s − 6·17-s + 4·19-s + 26-s − 6·29-s + 4·31-s + 32-s − 6·34-s + 10·37-s + 4·38-s + 6·41-s − 8·43-s + 52-s − 6·53-s − 6·58-s − 12·59-s − 14·61-s + 4·62-s + 64-s + 4·67-s − 6·68-s + 2·73-s + 10·74-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.196·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.21·43-s + 0.138·52-s − 0.824·53-s − 0.787·58-s − 1.56·59-s − 1.79·61-s + 0.508·62-s + 1/8·64-s + 0.488·67-s − 0.727·68-s + 0.234·73-s + 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.05957124006076, −12.60385517215265, −11.97566052039350, −11.67152234506112, −11.10585449256841, −10.88666432723297, −10.35517366835805, −9.659538278108796, −9.211148070048841, −9.023589757774243, −8.060764042754454, −7.844747884626273, −7.389798903488248, −6.592814152436214, −6.417886021101832, −5.906082230100062, −5.305946643489700, −4.731187208037989, −4.433072886077378, −3.818633726375058, −3.248817210449974, −2.763278852947059, −2.147476612201964, −1.586161746348575, −0.8655793790134818, 0,
0.8655793790134818, 1.586161746348575, 2.147476612201964, 2.763278852947059, 3.248817210449974, 3.818633726375058, 4.433072886077378, 4.731187208037989, 5.305946643489700, 5.906082230100062, 6.417886021101832, 6.592814152436214, 7.389798903488248, 7.844747884626273, 8.060764042754454, 9.023589757774243, 9.211148070048841, 9.659538278108796, 10.35517366835805, 10.88666432723297, 11.10585449256841, 11.67152234506112, 11.97566052039350, 12.60385517215265, 13.05957124006076