| L(s) = 1 | + 4·2-s + 8·4-s − 52·7-s − 116·11-s + 56·13-s − 208·14-s − 64·16-s − 464·22-s + 36·23-s + 224·26-s − 416·28-s − 256·32-s − 272·37-s − 476·41-s − 928·44-s + 144·46-s − 532·47-s + 1.35e3·49-s + 448·52-s + 1.01e3·53-s − 512·64-s − 1.08e3·74-s + 6.03e3·77-s − 729·81-s − 1.90e3·82-s − 2.91e3·91-s + 288·92-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 2.80·7-s − 3.17·11-s + 1.19·13-s − 3.97·14-s − 16-s − 4.49·22-s + 0.326·23-s + 1.68·26-s − 2.80·28-s − 1.41·32-s − 1.20·37-s − 1.81·41-s − 3.17·44-s + 0.461·46-s − 1.65·47-s + 3.94·49-s + 1.19·52-s + 2.63·53-s − 64-s − 1.70·74-s + 8.92·77-s − 81-s − 2.56·82-s − 3.35·91-s + 0.326·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4047673835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4047673835\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 52 T + 1352 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 56 T + 1568 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2158 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 272 T + 36992 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 238 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 532 T + 141512 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1016 T + 516128 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 340002 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 421902 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
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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
−12.43103872101815167118617711580, −12.14690215024153839073328862921, −11.38683608792695511211455436132, −10.60343477598138379602104910713, −10.48900422183040796128122459757, −9.890935023521847372805695133512, −9.518074779762942139254815373077, −8.558503727476825070030049891585, −8.411504276732927197082327066013, −7.38805120794804611820996825915, −6.84227511902004038581559717243, −6.50254427992281468767834767611, −5.70853957950394241867195164831, −5.52086502735337924280648565057, −4.89042703343782663207348278202, −3.87611490988332518231304816645, −3.20126252694756590445488499822, −3.07677245506884534954625107414, −2.31401891087876193663124348968, −0.20035381341151811075342004146,
0.20035381341151811075342004146, 2.31401891087876193663124348968, 3.07677245506884534954625107414, 3.20126252694756590445488499822, 3.87611490988332518231304816645, 4.89042703343782663207348278202, 5.52086502735337924280648565057, 5.70853957950394241867195164831, 6.50254427992281468767834767611, 6.84227511902004038581559717243, 7.38805120794804611820996825915, 8.411504276732927197082327066013, 8.558503727476825070030049891585, 9.518074779762942139254815373077, 9.890935023521847372805695133512, 10.48900422183040796128122459757, 10.60343477598138379602104910713, 11.38683608792695511211455436132, 12.14690215024153839073328862921, 12.43103872101815167118617711580
