| L(s) = 1 | − 16·4-s − 71·7-s + 337·13-s − 1.20e3·19-s − 625·25-s + 1.13e3·28-s − 194·31-s − 1.05e3·37-s + 3.21e3·43-s + 2.40e3·49-s − 5.39e3·52-s − 7.19e3·61-s + 4.09e3·64-s − 2.90e3·67-s − 2.49e3·73-s + 1.92e4·76-s − 4.67e3·79-s − 2.39e4·91-s − 9.07e3·97-s + 1.00e4·100-s + 1.98e4·103-s + 4.40e4·109-s − 1.46e4·121-s + 3.10e3·124-s + ⋯ |
| L(s) = 1 | − 4-s − 1.44·7-s + 1.99·13-s − 3.32·19-s − 25-s + 1.44·28-s − 0.201·31-s − 0.772·37-s + 1.73·43-s + 49-s − 1.99·52-s − 1.93·61-s + 64-s − 0.646·67-s − 0.468·73-s + 3.32·76-s − 0.749·79-s − 2.88·91-s − 0.964·97-s + 100-s + 1.87·103-s + 3.70·109-s − 121-s + 0.201·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5691743094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5691743094\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 23 T + p^{4} T^{2} )( 1 + 94 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 191 T + p^{4} T^{2} )( 1 - 146 T + p^{4} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 601 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 1559 T + p^{4} T^{2} )( 1 + 1753 T + p^{4} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 529 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3191 T + p^{4} T^{2} )( 1 - 23 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 1966 T + p^{4} T^{2} )( 1 + 5233 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5906 T + p^{4} T^{2} )( 1 + 8809 T + p^{4} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1249 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 7682 T + p^{4} T^{2} )( 1 + 12361 T + p^{4} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9743 T + p^{4} T^{2} )( 1 + 18814 T + p^{4} T^{2} ) \) |
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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
−13.94579870095200770565710636896, −13.09516018904253517801615138367, −12.98736411383849658028550935122, −12.64983328405788794463869124049, −11.76850859752348433369143610723, −10.93720707151669629777714345889, −10.60323299408741565936592400739, −10.05100088573048370879608789769, −9.212114724144471015767174281082, −8.807667928680595782478912815883, −8.563035203194021190923595709796, −7.64669056413070979617927312834, −6.57629322554324028343004282048, −6.25103316880298593900780465613, −5.73035025046356167552515514518, −4.30829255519721293153534732726, −4.12205400580966726825238146755, −3.22892814327337813813069071862, −1.95008840979031324587869547035, −0.37490054776353981793614398891,
0.37490054776353981793614398891, 1.95008840979031324587869547035, 3.22892814327337813813069071862, 4.12205400580966726825238146755, 4.30829255519721293153534732726, 5.73035025046356167552515514518, 6.25103316880298593900780465613, 6.57629322554324028343004282048, 7.64669056413070979617927312834, 8.563035203194021190923595709796, 8.807667928680595782478912815883, 9.212114724144471015767174281082, 10.05100088573048370879608789769, 10.60323299408741565936592400739, 10.93720707151669629777714345889, 11.76850859752348433369143610723, 12.64983328405788794463869124049, 12.98736411383849658028550935122, 13.09516018904253517801615138367, 13.94579870095200770565710636896
