| L(s) = 1 | + 4·2-s + 2·3-s − 4-s + 8·6-s − 2·7-s − 40·8-s − 48·9-s − 38·11-s − 2·12-s + 80·13-s − 8·14-s − 79·16-s + 34·17-s − 192·18-s − 180·19-s − 4·21-s − 152·22-s + 42·23-s − 80·24-s + 320·26-s − 146·27-s + 2·28-s − 216·29-s + 70·31-s + 76·32-s − 76·33-s + 136·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s − 1/8·4-s + 0.544·6-s − 0.107·7-s − 1.76·8-s − 1.77·9-s − 1.04·11-s − 0.0481·12-s + 1.70·13-s − 0.152·14-s − 1.23·16-s + 0.485·17-s − 2.51·18-s − 2.17·19-s − 0.0415·21-s − 1.47·22-s + 0.380·23-s − 0.680·24-s + 2.41·26-s − 1.04·27-s + 0.0134·28-s − 1.38·29-s + 0.405·31-s + 0.419·32-s − 0.400·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 17 T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 444 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 38 T + 2516 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 80 T + 3642 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 180 T + 21710 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 42 T - 5828 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 216 T + 54094 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 70 T + 1164 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 352 T + 98574 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 106934 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 88 T + 78282 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 348 T + 237730 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 292190 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1436 T + 913214 T^{2} + 1436 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 76 T + 239934 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 44 T + 317418 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1510 T + 1243364 T^{2} + 1510 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1088 T + 1072998 T^{2} + 1088 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 958 T + 642612 T^{2} + 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 248 T + 989018 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 204 T - 231350 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1012 T + 1511094 T^{2} + 1012 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.49456721504923835395791994559, −10.46939804110687834070385571951, −9.303866227542666238364532887991, −9.121853605285902917156136381891, −8.553843112070283183767884987293, −8.539657131233333845127812125973, −7.86251560995211461208681199342, −7.31635834503691947410243963895, −6.20675223507473312282609483555, −6.17369527762298940307162193000, −5.55369618065869437709346398265, −5.42188174292191170283196974704, −4.34344586341240930239854465656, −4.34090624199632028934399414609, −3.44584613170961419222844965534, −3.11095931456856298072474610661, −2.59790297775355187912239737067, −1.58787344218003049689483493107, 0, 0,
1.58787344218003049689483493107, 2.59790297775355187912239737067, 3.11095931456856298072474610661, 3.44584613170961419222844965534, 4.34090624199632028934399414609, 4.34344586341240930239854465656, 5.42188174292191170283196974704, 5.55369618065869437709346398265, 6.17369527762298940307162193000, 6.20675223507473312282609483555, 7.31635834503691947410243963895, 7.86251560995211461208681199342, 8.539657131233333845127812125973, 8.553843112070283183767884987293, 9.121853605285902917156136381891, 9.303866227542666238364532887991, 10.46939804110687834070385571951, 10.49456721504923835395791994559
