L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·7-s + 4·8-s − 4·10-s + 6·12-s − 8·13-s − 4·14-s − 4·15-s + 5·16-s − 8·17-s − 2·19-s − 6·20-s − 4·21-s + 6·23-s + 8·24-s + 3·25-s − 16·26-s − 2·27-s − 6·28-s − 2·29-s − 8·30-s + 8·31-s + 6·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s − 1.26·10-s + 1.73·12-s − 2.21·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 0.458·19-s − 1.34·20-s − 0.872·21-s + 1.25·23-s + 1.63·24-s + 3/5·25-s − 3.13·26-s − 0.384·27-s − 1.13·28-s − 0.371·29-s − 1.46·30-s + 1.43·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.614689192\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.614689192\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.70014702013236625185555076843, −7.63158793831171472169127950045, −7.07283322824950512924685272291, −6.91579256928465189083709205078, −6.77277104262888667624762237853, −6.35130601541978096524953519526, −5.73242876318444687771205710903, −5.52023223740105866872014185815, −4.90932730372051036655758177347, −4.81341498321746847261400073540, −4.36786582572724535628810358314, −4.10031726532696832066542211065, −3.61768679721791161998681006286, −3.40313782158905782420318751953, −2.73108234649569775123328808523, −2.61240597862726598612148924458, −2.29668255169620977761687983713, −2.13122510455035362697207961336, −0.959330441740456959268937777453, −0.41965663540441631266224656365,
0.41965663540441631266224656365, 0.959330441740456959268937777453, 2.13122510455035362697207961336, 2.29668255169620977761687983713, 2.61240597862726598612148924458, 2.73108234649569775123328808523, 3.40313782158905782420318751953, 3.61768679721791161998681006286, 4.10031726532696832066542211065, 4.36786582572724535628810358314, 4.81341498321746847261400073540, 4.90932730372051036655758177347, 5.52023223740105866872014185815, 5.73242876318444687771205710903, 6.35130601541978096524953519526, 6.77277104262888667624762237853, 6.91579256928465189083709205078, 7.07283322824950512924685272291, 7.63158793831171472169127950045, 7.70014702013236625185555076843