| L(s) = 1 | − 2·13-s + 2·17-s − 10·23-s + 13·25-s − 8·29-s − 6·43-s + 12·49-s + 6·53-s + 2·79-s − 32·101-s − 10·103-s − 14·107-s − 8·113-s + 23·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.554·13-s + 0.485·17-s − 2.08·23-s + 13/5·25-s − 1.48·29-s − 0.914·43-s + 12/7·49-s + 0.824·53-s + 0.225·79-s − 3.18·101-s − 0.985·103-s − 1.35·107-s − 0.752·113-s + 2.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.461524001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461524001\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 13 T^{2} + 83 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 97 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 23 T^{2} + 291 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 55 T^{2} + 1395 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 96 T^{2} + 4078 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 7057 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 3425 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 113 T^{2} + 7527 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 2153 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 93 T^{2} + 5359 T^{4} - 93 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 21 T^{2} + 5875 T^{4} - 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 149 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 35057 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 20 T^{2} - 5370 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 15606 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.86081308831566665720565259490, −6.71538229649236637221910496228, −6.52201945725062682748066029836, −5.95625987197202107559302074659, −5.95438124471610330306913205743, −5.66876690262703788335269341674, −5.62304903208554098870652527726, −5.36857789443278120414881660493, −5.04805061403019335681094120893, −4.83757946842746979179800288435, −4.52209860943223431301557428930, −4.39067461647639272501904916758, −4.03554564449177970218958763027, −4.02770735257650264101811393473, −3.46953691503004261543016016609, −3.41075256485519036425042778102, −3.14626544167937060921190277920, −2.77302871276348780070129146659, −2.54876915561826004222691273884, −2.12523712925333457415596365134, −2.06661315130203831413773451259, −1.62333519371937389405851323319, −1.18360499030640353242004599010, −0.868547254888743061771170972678, −0.26378253892600386308291257745,
0.26378253892600386308291257745, 0.868547254888743061771170972678, 1.18360499030640353242004599010, 1.62333519371937389405851323319, 2.06661315130203831413773451259, 2.12523712925333457415596365134, 2.54876915561826004222691273884, 2.77302871276348780070129146659, 3.14626544167937060921190277920, 3.41075256485519036425042778102, 3.46953691503004261543016016609, 4.02770735257650264101811393473, 4.03554564449177970218958763027, 4.39067461647639272501904916758, 4.52209860943223431301557428930, 4.83757946842746979179800288435, 5.04805061403019335681094120893, 5.36857789443278120414881660493, 5.62304903208554098870652527726, 5.66876690262703788335269341674, 5.95438124471610330306913205743, 5.95625987197202107559302074659, 6.52201945725062682748066029836, 6.71538229649236637221910496228, 6.86081308831566665720565259490
