| L(s) = 1 | − 2·4-s − 4·7-s − 4·9-s − 3·11-s + 4·13-s − 4·19-s − 3·23-s + 7·25-s + 8·28-s − 2·29-s − 7·31-s + 8·36-s + 5·37-s − 2·41-s − 8·43-s + 6·44-s + 3·47-s + 6·49-s − 8·52-s + 3·53-s + 16·59-s − 3·61-s + 16·63-s + 8·64-s − 8·67-s − 6·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s − 4/3·9-s − 0.904·11-s + 1.10·13-s − 0.917·19-s − 0.625·23-s + 7/5·25-s + 1.51·28-s − 0.371·29-s − 1.25·31-s + 4/3·36-s + 0.821·37-s − 0.312·41-s − 1.21·43-s + 0.904·44-s + 0.437·47-s + 6/7·49-s − 1.10·52-s + 0.412·53-s + 2.08·59-s − 0.384·61-s + 2.01·63-s + 64-s − 0.977·67-s − 0.712·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11179 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11179 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 1597 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 54 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 3 p T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T - 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 140 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 148 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−16.6538314609, −16.3060769100, −15.8394140908, −15.1677833723, −14.7056306356, −14.1817583541, −13.5944853042, −13.2314666215, −12.8246522679, −12.5452831699, −11.4442199418, −11.2910250104, −10.3344594540, −10.1894171311, −9.29135386588, −8.77398597037, −8.63529184736, −7.85436590865, −6.90726449978, −6.30769512609, −5.71271219400, −5.11392864557, −4.08560464628, −3.40279491410, −2.58581448484, 0,
2.58581448484, 3.40279491410, 4.08560464628, 5.11392864557, 5.71271219400, 6.30769512609, 6.90726449978, 7.85436590865, 8.63529184736, 8.77398597037, 9.29135386588, 10.1894171311, 10.3344594540, 11.2910250104, 11.4442199418, 12.5452831699, 12.8246522679, 13.2314666215, 13.5944853042, 14.1817583541, 14.7056306356, 15.1677833723, 15.8394140908, 16.3060769100, 16.6538314609
