L(s) = 1 | − 2-s + 3-s − 4-s − 4·5-s − 6-s + 3·8-s + 9-s + 4·10-s − 12-s − 4·15-s − 16-s − 18-s + 8·19-s + 4·20-s + 3·24-s + 2·25-s + 27-s − 4·29-s + 4·30-s − 5·32-s − 36-s − 8·38-s − 12·40-s − 8·43-s − 4·45-s − 48-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.03·15-s − 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.894·20-s + 0.612·24-s + 2/5·25-s + 0.192·27-s − 0.742·29-s + 0.730·30-s − 0.883·32-s − 1/6·36-s − 1.29·38-s − 1.89·40-s − 1.21·43-s − 0.596·45-s − 0.144·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6646449027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6646449027\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.724661210726173973340366981146, −9.042140676903253409137259496371, −8.672365196683144779961098572501, −8.229992153219430708166678625388, −7.70691183501011689426667743544, −7.32441936749744810989380005535, −7.25047783802838427330028450541, −6.14521652142582152085553341069, −5.34641745697560833577710874285, −4.80722452268725035282926398957, −4.13559084050773741974089362056, −3.62256199043071801163653681899, −3.22746506277116131252491518684, −1.93480858488759200478154145238, −0.69577924383102779401083388688,
0.69577924383102779401083388688, 1.93480858488759200478154145238, 3.22746506277116131252491518684, 3.62256199043071801163653681899, 4.13559084050773741974089362056, 4.80722452268725035282926398957, 5.34641745697560833577710874285, 6.14521652142582152085553341069, 7.25047783802838427330028450541, 7.32441936749744810989380005535, 7.70691183501011689426667743544, 8.229992153219430708166678625388, 8.672365196683144779961098572501, 9.042140676903253409137259496371, 9.724661210726173973340366981146