| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·11-s − 4·13-s + 5·16-s − 8·17-s − 4·19-s − 4·22-s − 8·26-s + 8·29-s − 12·31-s + 6·32-s − 16·34-s + 4·37-s − 8·38-s + 8·41-s − 6·44-s − 4·47-s − 12·52-s − 4·53-s + 16·58-s + 16·59-s − 20·61-s − 24·62-s + 7·64-s − 4·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s − 1.10·13-s + 5/4·16-s − 1.94·17-s − 0.917·19-s − 0.852·22-s − 1.56·26-s + 1.48·29-s − 2.15·31-s + 1.06·32-s − 2.74·34-s + 0.657·37-s − 1.29·38-s + 1.24·41-s − 0.904·44-s − 0.583·47-s − 1.66·52-s − 0.549·53-s + 2.10·58-s + 2.08·59-s − 2.56·61-s − 3.04·62-s + 7/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 160 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.40390696552429875805816987607, −7.07149514668228748995453113511, −6.78633192065727996855660760536, −6.40619557849196721234626637624, −6.05192671395370405034974781964, −5.88568236290735424417634580988, −5.30568208992729320133223596045, −5.07365129834296019028673126943, −4.57417664900218053085768901530, −4.55106686279006346469585476986, −4.03230845911111815865991516512, −3.87676902674504075288729074028, −3.13243545207358721453027083435, −2.80952576827936694236965068411, −2.51016169042402406257076724758, −2.20547063448227657421133644363, −1.71368723360904622860530757896, −1.19238049380801914993518122461, 0, 0,
1.19238049380801914993518122461, 1.71368723360904622860530757896, 2.20547063448227657421133644363, 2.51016169042402406257076724758, 2.80952576827936694236965068411, 3.13243545207358721453027083435, 3.87676902674504075288729074028, 4.03230845911111815865991516512, 4.55106686279006346469585476986, 4.57417664900218053085768901530, 5.07365129834296019028673126943, 5.30568208992729320133223596045, 5.88568236290735424417634580988, 6.05192671395370405034974781964, 6.40619557849196721234626637624, 6.78633192065727996855660760536, 7.07149514668228748995453113511, 7.40390696552429875805816987607
