| L(s) = 1 | + 4·5-s + 9-s − 2·13-s + 12·17-s + 2·25-s + 4·29-s + 20·37-s + 4·41-s + 4·45-s − 10·49-s − 4·53-s + 12·61-s − 8·65-s − 12·73-s + 81-s + 48·85-s + 20·89-s + 20·97-s + 12·101-s + 4·109-s + 28·113-s − 2·117-s − 18·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s − 0.554·13-s + 2.91·17-s + 2/5·25-s + 0.742·29-s + 3.28·37-s + 0.624·41-s + 0.596·45-s − 1.42·49-s − 0.549·53-s + 1.53·61-s − 0.992·65-s − 1.40·73-s + 1/9·81-s + 5.20·85-s + 2.11·89-s + 2.03·97-s + 1.19·101-s + 0.383·109-s + 2.63·113-s − 0.184·117-s − 1.63·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.010823441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.010823441\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.896842730015818559963999252704, −7.59875356765337238034292890321, −7.06541986235410893916933421520, −6.28159176985699903858961238086, −6.14841639240660146897531624692, −5.78517701336628451955041513916, −5.44020310407890724415031834900, −4.68403548258509043319774249783, −4.65680782039109928862677817630, −3.55810218913298118185059814514, −3.37296560686463484601752098732, −2.38879148145950189793255761663, −2.36636166197971073120579022543, −1.33803744783185283027867448010, −0.983919193523929230329764658751,
0.983919193523929230329764658751, 1.33803744783185283027867448010, 2.36636166197971073120579022543, 2.38879148145950189793255761663, 3.37296560686463484601752098732, 3.55810218913298118185059814514, 4.65680782039109928862677817630, 4.68403548258509043319774249783, 5.44020310407890724415031834900, 5.78517701336628451955041513916, 6.14841639240660146897531624692, 6.28159176985699903858961238086, 7.06541986235410893916933421520, 7.59875356765337238034292890321, 7.896842730015818559963999252704
