| L(s) = 1 | + 12·9-s − 8·41-s − 12·49-s + 90·81-s − 56·89-s − 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·9-s − 1.24·41-s − 1.71·49-s + 10·81-s − 5.93·89-s − 3.27·121-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 + 0.923·169-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 + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084842853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084842853\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 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
−6.36698517110010895693143815647, −5.89436892914822258699754208133, −5.61279634063926137773239755657, −5.57770383685539705263763834450, −5.30999175974862547841902186189, −4.98174706947258424915951031379, −4.80994258513261655632253113213, −4.80964536829037161654070140210, −4.40690917721682521284746137494, −4.26084194577213046080715369399, −4.10574934842512539013879318888, −3.98681574946037135357774559178, −3.69555219287335793129586724840, −3.55783863226895609980857454266, −3.17853737809823398660216616788, −2.92889693527705747627755365726, −2.77270984437400685218984501930, −2.38228239114615263251717266284, −1.98061311870984812950746836486, −1.81343494848902307280838838410, −1.67436883527775442014440289605, −1.33448767356358848593747205361, −1.06386010922047019019308053790, −1.00004350701790235168483505505, −0.12894277157611894695310244405,
0.12894277157611894695310244405, 1.00004350701790235168483505505, 1.06386010922047019019308053790, 1.33448767356358848593747205361, 1.67436883527775442014440289605, 1.81343494848902307280838838410, 1.98061311870984812950746836486, 2.38228239114615263251717266284, 2.77270984437400685218984501930, 2.92889693527705747627755365726, 3.17853737809823398660216616788, 3.55783863226895609980857454266, 3.69555219287335793129586724840, 3.98681574946037135357774559178, 4.10574934842512539013879318888, 4.26084194577213046080715369399, 4.40690917721682521284746137494, 4.80964536829037161654070140210, 4.80994258513261655632253113213, 4.98174706947258424915951031379, 5.30999175974862547841902186189, 5.57770383685539705263763834450, 5.61279634063926137773239755657, 5.89436892914822258699754208133, 6.36698517110010895693143815647
