| L(s) = 1 | + 3-s + 6·5-s − 7-s + 6·11-s + 6·15-s − 5·19-s − 21-s − 12·23-s + 19·25-s − 27-s + 5·31-s + 6·33-s − 6·35-s + 11·37-s + 6·47-s − 6·49-s − 12·53-s + 36·55-s − 5·57-s − 12·59-s − 24·61-s − 15·67-s − 12·69-s − 9·73-s + 19·75-s − 6·77-s − 21·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.68·5-s − 0.377·7-s + 1.80·11-s + 1.54·15-s − 1.14·19-s − 0.218·21-s − 2.50·23-s + 19/5·25-s − 0.192·27-s + 0.898·31-s + 1.04·33-s − 1.01·35-s + 1.80·37-s + 0.875·47-s − 6/7·49-s − 1.64·53-s + 4.85·55-s − 0.662·57-s − 1.56·59-s − 3.07·61-s − 1.83·67-s − 1.44·69-s − 1.05·73-s + 2.19·75-s − 0.683·77-s − 2.36·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.072855291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072855291\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−11.83959352537171871881558092675, −11.34867077618516947519672883509, −10.47641176758024532273549197545, −10.31505460159617154616572448919, −9.927871008621372283969602023539, −9.275575648154472178810909426504, −9.109433634705778217341884943107, −9.008654293752635906015960744076, −7.88442733305591878489796024493, −7.76890664811245694987480613188, −6.52643661710426782187699777452, −6.33809972791854056707354639362, −6.07341180484282196462216718502, −5.81035001421093663734180988720, −4.53932921393914051605084982138, −4.42272945846061880247644676667, −3.35989489619348189556796851941, −2.65738939984752821013330157489, −1.87462165767029376235891253597, −1.58199541757540501807887980487,
1.58199541757540501807887980487, 1.87462165767029376235891253597, 2.65738939984752821013330157489, 3.35989489619348189556796851941, 4.42272945846061880247644676667, 4.53932921393914051605084982138, 5.81035001421093663734180988720, 6.07341180484282196462216718502, 6.33809972791854056707354639362, 6.52643661710426782187699777452, 7.76890664811245694987480613188, 7.88442733305591878489796024493, 9.008654293752635906015960744076, 9.109433634705778217341884943107, 9.275575648154472178810909426504, 9.927871008621372283969602023539, 10.31505460159617154616572448919, 10.47641176758024532273549197545, 11.34867077618516947519672883509, 11.83959352537171871881558092675
