| L(s) = 1 | + 2·3-s − 2·7-s + 5·9-s + 2·11-s − 4·13-s + 4·17-s − 8·19-s − 4·21-s + 8·23-s + 8·25-s + 10·27-s + 4·29-s − 8·31-s + 4·33-s + 4·37-s − 8·39-s + 8·41-s + 32·43-s + 4·47-s + 7·49-s + 8·51-s − 8·53-s − 16·57-s + 6·59-s + 22·61-s − 10·63-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 5/3·9-s + 0.603·11-s − 1.10·13-s + 0.970·17-s − 1.83·19-s − 0.872·21-s + 1.66·23-s + 8/5·25-s + 1.92·27-s + 0.742·29-s − 1.43·31-s + 0.696·33-s + 0.657·37-s − 1.28·39-s + 1.24·41-s + 4.87·43-s + 0.583·47-s + 49-s + 1.12·51-s − 1.09·53-s − 2.11·57-s + 0.781·59-s + 2.81·61-s − 1.25·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.545714148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.545714148\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 12 T^{2} + 112 T^{3} + 1127 T^{4} + 112 p T^{5} + 12 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 4 T^{2} - 112 T^{3} + 1599 T^{4} - 112 p T^{5} + 4 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 128 T^{3} - 829 T^{4} - 128 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 184 T^{3} - 1177 T^{4} + 184 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 74 T^{2} + 16 T^{3} + 5139 T^{4} + 16 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 56 T^{2} + 112 T^{3} + 8199 T^{4} + 112 p T^{5} - 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 89 T^{2} - 42 T^{3} + 10020 T^{4} - 42 p T^{5} - 89 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 22 T + 249 T^{2} - 2486 T^{3} + 21980 T^{4} - 2486 p T^{5} + 249 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 81 T^{2} + 98 T^{3} + 2468 T^{4} + 98 p T^{5} - 81 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 224 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 132 T^{2} - 8 T^{3} + 15407 T^{4} - 8 p T^{5} - 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T - 105 T^{2} + 98 T^{3} + 5324 T^{4} + 98 p T^{5} - 105 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 262 T^{2} - 3264 T^{3} + 39411 T^{4} - 3264 p T^{5} + 262 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.07419566056211117856673639078, −6.56792313328835432616465678396, −6.55092074386906787347873162157, −6.49507146445910643177311459512, −6.39388885574494442309311752717, −5.71453225292799946193955426103, −5.49108491108645461903257310658, −5.35913816293157517649850798414, −5.27921847385394017735834954711, −4.83901836703066958258953984551, −4.67838613282785002363249699032, −4.22185879222684714702754044908, −4.08882095800274966357314235915, −3.95674420380050696630317704880, −3.70897849262593270327207946083, −3.64478851050808857923622297460, −2.85421765224126827791523280846, −2.82957817773784741283924099015, −2.66307130587684243076395244235, −2.39748598588388749359582522728, −2.18627785894050671913497548053, −1.75935902826918923463083187250, −1.04742463516240838060276311187, −0.879355077469824128975951296143, −0.73912507863496100969202423434,
0.73912507863496100969202423434, 0.879355077469824128975951296143, 1.04742463516240838060276311187, 1.75935902826918923463083187250, 2.18627785894050671913497548053, 2.39748598588388749359582522728, 2.66307130587684243076395244235, 2.82957817773784741283924099015, 2.85421765224126827791523280846, 3.64478851050808857923622297460, 3.70897849262593270327207946083, 3.95674420380050696630317704880, 4.08882095800274966357314235915, 4.22185879222684714702754044908, 4.67838613282785002363249699032, 4.83901836703066958258953984551, 5.27921847385394017735834954711, 5.35913816293157517649850798414, 5.49108491108645461903257310658, 5.71453225292799946193955426103, 6.39388885574494442309311752717, 6.49507146445910643177311459512, 6.55092074386906787347873162157, 6.56792313328835432616465678396, 7.07419566056211117856673639078
