| L(s) = 1 | − 4·2-s − 3·3-s + 8·4-s − 4·5-s + 12·6-s − 32·8-s + 16·10-s − 62·11-s − 24·12-s + 124·13-s + 12·15-s + 128·16-s + 84·17-s + 100·19-s − 32·20-s + 248·22-s + 42·23-s + 96·24-s + 125·25-s − 496·26-s + 27·27-s − 20·29-s − 48·30-s − 48·31-s − 256·32-s + 186·33-s − 336·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.357·5-s + 0.816·6-s − 1.41·8-s + 0.505·10-s − 1.69·11-s − 0.577·12-s + 2.64·13-s + 0.206·15-s + 2·16-s + 1.19·17-s + 1.20·19-s − 0.357·20-s + 2.40·22-s + 0.380·23-s + 0.816·24-s + 25-s − 3.74·26-s + 0.192·27-s − 0.128·29-s − 0.292·30-s − 0.278·31-s − 1.41·32-s + 0.981·33-s − 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7272198123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7272198123\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 62 T + 2513 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 100 T + 3141 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T - 10403 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 48 T - 27487 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 248 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 324 T + 1153 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 258 T - 82313 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 120 T - 190979 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 622 T + 159903 T^{2} - 622 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 904 T + 516453 T^{2} + 904 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 678 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 642 T + 23147 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 740 T + 54561 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 200 T - 664969 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1266 T + p^{3} T^{2} )^{2} \) |
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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
−12.85479486593872332891213667203, −12.20654698366684309537075686440, −11.44755393789103818578019655473, −11.35968158873318476621115788050, −10.66414978980226135028134949260, −10.38013495944352052875608792251, −9.810059897962084587013298135216, −9.038279889944003102409311114258, −8.781185927318730634156833235090, −8.243925448836237897780838406800, −7.57678477131515501916379363705, −7.39098578178639916165874534573, −6.22011346981919278764205532236, −5.76817100537279140290380582329, −5.54772832288477779237806709428, −4.34187355624120013178779368593, −3.24179831843741899440201963294, −2.90239732560987815559884336063, −1.17608250710157843164325658666, −0.66685239946383197804067101114,
0.66685239946383197804067101114, 1.17608250710157843164325658666, 2.90239732560987815559884336063, 3.24179831843741899440201963294, 4.34187355624120013178779368593, 5.54772832288477779237806709428, 5.76817100537279140290380582329, 6.22011346981919278764205532236, 7.39098578178639916165874534573, 7.57678477131515501916379363705, 8.243925448836237897780838406800, 8.781185927318730634156833235090, 9.038279889944003102409311114258, 9.810059897962084587013298135216, 10.38013495944352052875608792251, 10.66414978980226135028134949260, 11.35968158873318476621115788050, 11.44755393789103818578019655473, 12.20654698366684309537075686440, 12.85479486593872332891213667203
