| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 6·9-s + 10·11-s − 4·16-s + 6·17-s − 24·18-s − 6·19-s + 40·22-s + 7·25-s − 32·32-s + 24·34-s − 48·36-s − 24·38-s + 42·41-s − 18·43-s + 80·44-s − 2·49-s + 28·50-s − 64·64-s − 32·67-s + 48·68-s − 48·72-s − 30·73-s − 48·76-s + 27·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 2.82·8-s − 2·9-s + 3.01·11-s − 16-s + 1.45·17-s − 5.65·18-s − 1.37·19-s + 8.52·22-s + 7/5·25-s − 5.65·32-s + 4.11·34-s − 8·36-s − 3.89·38-s + 6.55·41-s − 2.74·43-s + 12.0·44-s − 2/7·49-s + 3.95·50-s − 8·64-s − 3.90·67-s + 5.82·68-s − 5.65·72-s − 3.51·73-s − 5.50·76-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.15824908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.15824908\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 9 T^{2} - 760 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - 21 T + 188 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 15 T^{2} - 2584 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 27 T + 326 T^{2} - 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64287110195845455600571975348, −7.60362909598465610715550719740, −7.38090039808502827257865916177, −6.85588749814962395249424105277, −6.54299872048024963407176290608, −6.53310558806318440545704429545, −6.30093137896911086764559836516, −6.04266861940329737826507298603, −5.72991353229310829738351881843, −5.71248863452681977192840752830, −5.71144359971032062863901817971, −4.79828964259493927283058134538, −4.78228879906470452476985319236, −4.66201839809633003629275355393, −4.44977000388254875768371618797, −3.95574820829292894217575661107, −3.71683860070410738945971181652, −3.55249505503508545433117141973, −3.30530371173192085313716048738, −2.91737771817043246453735946834, −2.72160982628742062208172612566, −2.31304894017500441067379821647, −1.94873601377448382464476098208, −1.27609857696236131307261105900, −0.70012247849513531544941636128,
0.70012247849513531544941636128, 1.27609857696236131307261105900, 1.94873601377448382464476098208, 2.31304894017500441067379821647, 2.72160982628742062208172612566, 2.91737771817043246453735946834, 3.30530371173192085313716048738, 3.55249505503508545433117141973, 3.71683860070410738945971181652, 3.95574820829292894217575661107, 4.44977000388254875768371618797, 4.66201839809633003629275355393, 4.78228879906470452476985319236, 4.79828964259493927283058134538, 5.71144359971032062863901817971, 5.71248863452681977192840752830, 5.72991353229310829738351881843, 6.04266861940329737826507298603, 6.30093137896911086764559836516, 6.53310558806318440545704429545, 6.54299872048024963407176290608, 6.85588749814962395249424105277, 7.38090039808502827257865916177, 7.60362909598465610715550719740, 7.64287110195845455600571975348
