| L(s) = 1 | − 2·3-s + 3·9-s + 6·13-s − 4·17-s + 8·23-s + 6·25-s − 4·27-s + 20·29-s − 12·39-s − 8·43-s + 10·49-s + 8·51-s + 12·53-s − 4·61-s − 16·69-s − 12·75-s + 5·81-s − 40·87-s − 4·101-s − 32·103-s − 16·107-s + 28·113-s + 18·117-s + 22·121-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.66·13-s − 0.970·17-s + 1.66·23-s + 6/5·25-s − 0.769·27-s + 3.71·29-s − 1.92·39-s − 1.21·43-s + 10/7·49-s + 1.12·51-s + 1.64·53-s − 0.512·61-s − 1.92·69-s − 1.38·75-s + 5/9·81-s − 4.28·87-s − 0.398·101-s − 3.15·103-s − 1.54·107-s + 2.63·113-s + 1.66·117-s + 2·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.405875264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.405875264\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.945788036158832306267704992298, −8.622705684907070998753851757528, −8.367517411383458877498500146946, −8.314867923470759919370174204799, −7.25906597754550183834240310899, −7.10921587357757010295235799710, −6.62293506178312334438084101834, −6.59696483526209601878506200631, −6.02439281888965314938366131321, −5.67672145735438302741027594839, −5.13575354460651775619037103489, −4.83787937079836053884547096337, −4.30043225343445173569649623752, −4.22500968495076618686509278589, −3.27904487397404063696102251563, −3.04904025041904822423244154549, −2.46411762025911602682651204632, −1.63894762662295027536884512983, −0.862667892041580733144885697872, −0.843748852506503738779317108194,
0.843748852506503738779317108194, 0.862667892041580733144885697872, 1.63894762662295027536884512983, 2.46411762025911602682651204632, 3.04904025041904822423244154549, 3.27904487397404063696102251563, 4.22500968495076618686509278589, 4.30043225343445173569649623752, 4.83787937079836053884547096337, 5.13575354460651775619037103489, 5.67672145735438302741027594839, 6.02439281888965314938366131321, 6.59696483526209601878506200631, 6.62293506178312334438084101834, 7.10921587357757010295235799710, 7.25906597754550183834240310899, 8.314867923470759919370174204799, 8.367517411383458877498500146946, 8.622705684907070998753851757528, 8.945788036158832306267704992298
