| L(s) = 1 | − 8·19-s − 10·25-s + 16·31-s + 28·49-s − 4·61-s − 32·79-s − 28·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 1.83·19-s − 2·25-s + 2.87·31-s + 4·49-s − 0.512·61-s − 3.60·79-s − 2.68·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7752095040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7752095040\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 14 T^{2} - 93 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 86 T^{2} + 4587 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.38202947985952455331813412350, −6.24338527266760267319667286264, −6.04585073767139421091738373249, −5.75694010282644296356819880689, −5.73791444505866388055019523154, −5.46968143340456393782470328339, −5.19358487080729097839251856381, −5.05242441455579779834229351709, −4.59411762949536955101042818879, −4.45646348502415518030999229896, −4.21989337670795579039681759743, −4.15016516307278174501134584781, −3.95817035018831696407019637069, −3.75918759375632040845616557223, −3.44457491308628533649228386600, −2.92925564378062650246976573786, −2.68991472823303391582786612582, −2.63303158490606636621750028848, −2.58450369575947933647075323165, −1.97855691606459226176201303408, −1.79989341853680827774497589699, −1.55754823870179099299157319695, −1.00891017164245261018619872863, −0.807614761992021626550787113733, −0.16227331582877620967493007376,
0.16227331582877620967493007376, 0.807614761992021626550787113733, 1.00891017164245261018619872863, 1.55754823870179099299157319695, 1.79989341853680827774497589699, 1.97855691606459226176201303408, 2.58450369575947933647075323165, 2.63303158490606636621750028848, 2.68991472823303391582786612582, 2.92925564378062650246976573786, 3.44457491308628533649228386600, 3.75918759375632040845616557223, 3.95817035018831696407019637069, 4.15016516307278174501134584781, 4.21989337670795579039681759743, 4.45646348502415518030999229896, 4.59411762949536955101042818879, 5.05242441455579779834229351709, 5.19358487080729097839251856381, 5.46968143340456393782470328339, 5.73791444505866388055019523154, 5.75694010282644296356819880689, 6.04585073767139421091738373249, 6.24338527266760267319667286264, 6.38202947985952455331813412350
