| L(s) = 1 | − 8·9-s − 8·13-s + 8·37-s − 32·41-s − 8·49-s + 8·53-s + 30·81-s + 24·89-s + 64·117-s − 20·121-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 + ⋯ |
| L(s) = 1 | − 8/3·9-s − 2.21·13-s + 1.31·37-s − 4.99·41-s − 8/7·49-s + 1.09·53-s + 10/3·81-s + 2.54·89-s + 5.91·117-s − 1.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7792892518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7792892518\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.08908857462758199892468837793, −6.07122527898236167442688338020, −5.79836908330729837949274552652, −5.28169079227228822184655948533, −5.21211786172296417949901974329, −5.16214678653234380312878798541, −5.07303301412199841480781907130, −4.96057225008866568474642554658, −4.48449722759761663254132003071, −4.44251298507713104343141950704, −4.06960217301759343924509923632, −3.62485130918037994995132472084, −3.54423479194525388847170848966, −3.51004291827364760066328430634, −3.14577901283815069531045905981, −2.81151172318587781076890306636, −2.66528146503537979315023091741, −2.50283260455491166639476772092, −2.42876500413008220011429506859, −1.89035277069802775473803596217, −1.68938267814546071963517182339, −1.53288075109768247654198184965, −0.874179073735913092808819460420, −0.33472462835483728800459642227, −0.29593623331808071186782751564,
0.29593623331808071186782751564, 0.33472462835483728800459642227, 0.874179073735913092808819460420, 1.53288075109768247654198184965, 1.68938267814546071963517182339, 1.89035277069802775473803596217, 2.42876500413008220011429506859, 2.50283260455491166639476772092, 2.66528146503537979315023091741, 2.81151172318587781076890306636, 3.14577901283815069531045905981, 3.51004291827364760066328430634, 3.54423479194525388847170848966, 3.62485130918037994995132472084, 4.06960217301759343924509923632, 4.44251298507713104343141950704, 4.48449722759761663254132003071, 4.96057225008866568474642554658, 5.07303301412199841480781907130, 5.16214678653234380312878798541, 5.21211786172296417949901974329, 5.28169079227228822184655948533, 5.79836908330729837949274552652, 6.07122527898236167442688338020, 6.08908857462758199892468837793
