L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 2·11-s + 4·13-s + 4·14-s + 5·16-s − 8·17-s + 4·19-s − 4·22-s − 8·26-s − 6·28-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 + 3·49-s + 12·52-s + 4·53-s + 8·56-s + 16·58-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s − 1.94·17-s + 0.917·19-s − 0.852·22-s − 1.56·26-s − 1.13·28-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 + 3/7·49-s + 1.66·52-s + 0.549·53-s + 1.06·56-s + 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.117852568\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117852568\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 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
−9.568720694492842332285005735232, −9.479154363203128727447259177332, −9.000573323073811753603733064343, −8.633398914850368824601514782239, −8.169029900938253929981344976643, −8.076332270978591892969200948968, −7.29101283497002339709785263193, −6.93511609982213879126543912391, −6.55564655002524323292668834712, −6.45218908251651398025115597534, −5.66008896736701228876561286618, −5.56084083773721755730789320532, −4.60639809766414292694415672538, −4.10819024174616492266277601488, −3.65288355157941774785309992502, −3.13889997743998876238520816209, −2.27965959149649790550620084538, −2.20906621040382106957390401019, −1.07450384758685947370688383531, −0.65731802928969972534507916828,
0.65731802928969972534507916828, 1.07450384758685947370688383531, 2.20906621040382106957390401019, 2.27965959149649790550620084538, 3.13889997743998876238520816209, 3.65288355157941774785309992502, 4.10819024174616492266277601488, 4.60639809766414292694415672538, 5.56084083773721755730789320532, 5.66008896736701228876561286618, 6.45218908251651398025115597534, 6.55564655002524323292668834712, 6.93511609982213879126543912391, 7.29101283497002339709785263193, 8.076332270978591892969200948968, 8.169029900938253929981344976643, 8.633398914850368824601514782239, 9.000573323073811753603733064343, 9.479154363203128727447259177332, 9.568720694492842332285005735232