| L(s) = 1 | − 2·3-s + 3·9-s − 4·17-s − 16·23-s + 6·25-s − 4·27-s − 4·29-s + 8·43-s − 2·49-s + 8·51-s + 12·53-s − 4·61-s + 32·69-s − 12·75-s − 16·79-s + 5·81-s + 8·87-s − 12·101-s − 16·103-s + 8·107-s + 36·113-s + 22·121-s + 127-s − 16·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 0.970·17-s − 3.33·23-s + 6/5·25-s − 0.769·27-s − 0.742·29-s + 1.21·43-s − 2/7·49-s + 1.12·51-s + 1.64·53-s − 0.512·61-s + 3.85·69-s − 1.38·75-s − 1.80·79-s + 5/9·81-s + 0.857·87-s − 1.19·101-s − 1.57·103-s + 0.773·107-s + 3.38·113-s + 2·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5591391160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5591391160\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.469406958442807821428975525692, −8.320369693221325341027655802851, −7.898380856199665783296892544218, −7.35016765942482375773154320930, −7.09998081692499823636777263886, −6.86340162247837341881340829021, −6.27214961142902803845116249639, −5.96440628541283493888670961192, −5.64833471268085440419633784565, −5.59789882671124399712988560726, −4.63953879852879865856699751452, −4.56949441544393243868015101293, −4.25091156028158522063167245238, −3.69947440476043309253858391302, −3.35156696219889861450894455035, −2.50720558585888979910815343873, −2.17603307483345049820413325202, −1.72434240454804191123836566322, −1.00412164802975041055540353189, −0.26360063506803633428966836616,
0.26360063506803633428966836616, 1.00412164802975041055540353189, 1.72434240454804191123836566322, 2.17603307483345049820413325202, 2.50720558585888979910815343873, 3.35156696219889861450894455035, 3.69947440476043309253858391302, 4.25091156028158522063167245238, 4.56949441544393243868015101293, 4.63953879852879865856699751452, 5.59789882671124399712988560726, 5.64833471268085440419633784565, 5.96440628541283493888670961192, 6.27214961142902803845116249639, 6.86340162247837341881340829021, 7.09998081692499823636777263886, 7.35016765942482375773154320930, 7.898380856199665783296892544218, 8.320369693221325341027655802851, 8.469406958442807821428975525692
