| L(s) = 1 | − 32·11-s + 24·17-s + 96·19-s − 4·25-s − 88·41-s − 160·43-s + 68·49-s − 128·59-s − 128·67-s + 120·73-s − 224·83-s − 312·89-s − 248·97-s − 192·107-s − 328·113-s + 252·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 92·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.90·11-s + 1.41·17-s + 5.05·19-s − 0.159·25-s − 2.14·41-s − 3.72·43-s + 1.38·49-s − 2.16·59-s − 1.91·67-s + 1.64·73-s − 2.69·83-s − 3.50·89-s − 2.55·97-s − 1.79·107-s − 2.90·113-s + 2.08·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.544·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1262625726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1262625726\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 486 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2886 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 16 T + 258 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 92 T^{2} - 17562 T^{4} - 92 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 422 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 48 T + 1250 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 994 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3068 T^{2} + 3748518 T^{4} - 3068 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3292 T^{2} + 3990 p^{2} T^{4} - 3292 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 44 T + 3654 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 80 T + 4098 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2180 T^{2} + 8539014 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11068 T^{2} + 46399206 T^{4} - 11068 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 64 T + 2178 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7246 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 64 T + 7650 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 60 T + 8486 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 112 T + 16866 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 156 T + 21158 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 124 T + 15750 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.04210695202699525203981849797, −5.71611470275304737213454745951, −5.65071406152909454608340190516, −5.44451156104615134181413449917, −5.43242587593914861997942320464, −5.24390863110268170509398236268, −5.18854356762923000249755998877, −4.86518821635856353544940135363, −4.58233623463315519606493747368, −4.25969409547056672157630634455, −4.25292037536988507365301003005, −3.55532674674327245243394971705, −3.36540487703239381113515927384, −3.35184001112761191888020731096, −3.23561570943049891956125350967, −2.82583631836625586953611203762, −2.81482336110461661548826111365, −2.53822157486449683859468043156, −2.20914079603164346614666299011, −1.57986694202122553515310138406, −1.32950494704440607447298536063, −1.25807467975227561691568413115, −1.25757110853355560600988955333, −0.37431713698897301608640160536, −0.06039082645072214260553643347,
0.06039082645072214260553643347, 0.37431713698897301608640160536, 1.25757110853355560600988955333, 1.25807467975227561691568413115, 1.32950494704440607447298536063, 1.57986694202122553515310138406, 2.20914079603164346614666299011, 2.53822157486449683859468043156, 2.81482336110461661548826111365, 2.82583631836625586953611203762, 3.23561570943049891956125350967, 3.35184001112761191888020731096, 3.36540487703239381113515927384, 3.55532674674327245243394971705, 4.25292037536988507365301003005, 4.25969409547056672157630634455, 4.58233623463315519606493747368, 4.86518821635856353544940135363, 5.18854356762923000249755998877, 5.24390863110268170509398236268, 5.43242587593914861997942320464, 5.44451156104615134181413449917, 5.65071406152909454608340190516, 5.71611470275304737213454745951, 6.04210695202699525203981849797
