| L(s) = 1 | − 4·3-s + 6·9-s + 24·19-s − 4·25-s + 4·27-s + 8·43-s + 20·49-s − 96·57-s − 8·67-s + 24·73-s + 16·75-s − 37·81-s + 40·97-s + 28·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 80·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 144·171-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 5.50·19-s − 4/5·25-s + 0.769·27-s + 1.21·43-s + 20/7·49-s − 12.7·57-s − 0.977·67-s + 2.80·73-s + 1.84·75-s − 4.11·81-s + 4.06·97-s + 2.54·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.59·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 11.0·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799986447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799986447\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−7.44136780119340210157190534445, −7.21429759978967445603884307278, −6.98618684393061331716006404217, −6.74912287965975433785368990004, −6.41597717823481055078715679937, −5.99766610609118302943099009075, −5.84741892368229345307279839219, −5.78703279602356452623271551160, −5.71894897693456412260101745725, −5.27160400599861948464020251522, −5.09992699319263516767130241532, −5.01255937581447671062533828346, −4.76249596032280168370694187900, −4.42019268857796413045899048568, −4.00163574267424099965367794325, −3.78078906294554177843693138900, −3.37628091851646804423964913142, −3.15473104332175028005239317372, −2.97702873101470854607841532407, −2.60993133914374800312022424799, −2.04296572744955548990160430870, −1.71617496885459994640486993606, −0.898916531007710969705436753683, −0.831482585609022497589204033973, −0.74608280365055639237827980187,
0.74608280365055639237827980187, 0.831482585609022497589204033973, 0.898916531007710969705436753683, 1.71617496885459994640486993606, 2.04296572744955548990160430870, 2.60993133914374800312022424799, 2.97702873101470854607841532407, 3.15473104332175028005239317372, 3.37628091851646804423964913142, 3.78078906294554177843693138900, 4.00163574267424099965367794325, 4.42019268857796413045899048568, 4.76249596032280168370694187900, 5.01255937581447671062533828346, 5.09992699319263516767130241532, 5.27160400599861948464020251522, 5.71894897693456412260101745725, 5.78703279602356452623271551160, 5.84741892368229345307279839219, 5.99766610609118302943099009075, 6.41597717823481055078715679937, 6.74912287965975433785368990004, 6.98618684393061331716006404217, 7.21429759978967445603884307278, 7.44136780119340210157190534445
