| L(s) = 1 | − 4·5-s + 2·9-s + 16·19-s + 2·25-s − 24·29-s − 8·45-s − 12·49-s + 64·71-s − 5·81-s − 64·95-s + 24·101-s + 28·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 32·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2/3·9-s + 3.67·19-s + 2/5·25-s − 4.45·29-s − 1.19·45-s − 1.71·49-s + 7.59·71-s − 5/9·81-s − 6.56·95-s + 2.38·101-s + 2.54·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 2.44·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.458339760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458339760\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−6.66019070988942103209521526539, −6.60769357655409187209813081700, −5.93074452427860483253368591776, −5.65419727021888271968029737796, −5.62483518300537895055594836450, −5.61133007659446458088482180396, −5.47332343351147726056863400261, −4.91293146761312859341446384146, −4.82065775456169383337557366706, −4.58369466921874490686803465310, −4.45419421535759320349314777968, −4.10307948658586372432469485547, −3.68196287965442043223861882673, −3.53463126884422926013912949084, −3.43656457884212122052744941585, −3.41874281992089013621927268304, −3.34402979999119092544758786131, −2.73183896855265236728220926973, −2.21784534978048113721821761327, −2.04304702867721560509611759075, −1.90091536429102656939489293854, −1.51303538510793581079964824895, −0.973697522167034719806590298320, −0.67083291082030328711885474432, −0.40079103931788379603632072418,
0.40079103931788379603632072418, 0.67083291082030328711885474432, 0.973697522167034719806590298320, 1.51303538510793581079964824895, 1.90091536429102656939489293854, 2.04304702867721560509611759075, 2.21784534978048113721821761327, 2.73183896855265236728220926973, 3.34402979999119092544758786131, 3.41874281992089013621927268304, 3.43656457884212122052744941585, 3.53463126884422926013912949084, 3.68196287965442043223861882673, 4.10307948658586372432469485547, 4.45419421535759320349314777968, 4.58369466921874490686803465310, 4.82065775456169383337557366706, 4.91293146761312859341446384146, 5.47332343351147726056863400261, 5.61133007659446458088482180396, 5.62483518300537895055594836450, 5.65419727021888271968029737796, 5.93074452427860483253368591776, 6.60769357655409187209813081700, 6.66019070988942103209521526539
