| L(s) = 1 | + 5-s − 6·7-s + 12·9-s + 16·23-s + 5·25-s − 6·35-s + 2·43-s + 12·45-s − 26·47-s + 23·49-s + 30·61-s − 72·63-s + 90·81-s + 64·83-s + 40·101-s + 16·115-s − 3·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 96·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2.26·7-s + 4·9-s + 3.33·23-s + 25-s − 1.01·35-s + 0.304·43-s + 1.78·45-s − 3.79·47-s + 23/7·49-s + 3.84·61-s − 9.07·63-s + 10·81-s + 7.02·83-s + 3.98·101-s + 1.49·115-s − 0.272·121-s + 1.25·125-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 − 7.56·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.303701719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.303701719\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} )( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 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
−6.86484106580748464882766542627, −6.44197683101117964505117265266, −6.36796656931150085371164308062, −6.33648322847351524664159744575, −6.23142674310024969192902607324, −5.69558695238981603077550187545, −5.20088839975652228743514485666, −5.03811106678401490139740864881, −5.01967165038520020020600042291, −4.86976979669395095926094719071, −4.57965804675782730634782857793, −4.41582660045873557184893338456, −3.93439653555148900325034012948, −3.59129639465755326704929307529, −3.50547817389557688949944355541, −3.42722780515022407641659120709, −3.39310593923467888935110964938, −2.74179374201405153941427138478, −2.38043669621356372176365064881, −2.19458815306264168957882313035, −2.00952701995217277751861758974, −1.32846950626729400691445266186, −1.24498120177484434051810935034, −0.74867003505030870655257915194, −0.70861126742914874651406108496,
0.70861126742914874651406108496, 0.74867003505030870655257915194, 1.24498120177484434051810935034, 1.32846950626729400691445266186, 2.00952701995217277751861758974, 2.19458815306264168957882313035, 2.38043669621356372176365064881, 2.74179374201405153941427138478, 3.39310593923467888935110964938, 3.42722780515022407641659120709, 3.50547817389557688949944355541, 3.59129639465755326704929307529, 3.93439653555148900325034012948, 4.41582660045873557184893338456, 4.57965804675782730634782857793, 4.86976979669395095926094719071, 5.01967165038520020020600042291, 5.03811106678401490139740864881, 5.20088839975652228743514485666, 5.69558695238981603077550187545, 6.23142674310024969192902607324, 6.33648322847351524664159744575, 6.36796656931150085371164308062, 6.44197683101117964505117265266, 6.86484106580748464882766542627
