| L(s) = 1 | + 3·2-s + 3-s + 7·4-s + 5·5-s + 3·6-s + 3·7-s + 15·8-s + 15·10-s − 11-s + 7·12-s + 4·13-s + 9·14-s + 5·15-s + 30·16-s − 12·17-s + 5·19-s + 35·20-s + 3·21-s − 3·22-s − 6·23-s + 15·24-s + 10·25-s + 12·26-s + 21·28-s + 20·29-s + 15·30-s + 13·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 7/2·4-s + 2.23·5-s + 1.22·6-s + 1.13·7-s + 5.30·8-s + 4.74·10-s − 0.301·11-s + 2.02·12-s + 1.10·13-s + 2.40·14-s + 1.29·15-s + 15/2·16-s − 2.91·17-s + 1.14·19-s + 7.82·20-s + 0.654·21-s − 0.639·22-s − 1.25·23-s + 3.06·24-s + 2·25-s + 2.35·26-s + 3.96·28-s + 3.71·29-s + 2.73·30-s + 2.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(47.28977683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(47.28977683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 5 T - 9 T^{2} + 5 p T^{3} - 184 T^{4} + 5 p^{2} T^{5} - 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 6 T + 13 T^{2} + 150 T^{3} + 1231 T^{4} + 150 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 20 T + 161 T^{2} - 680 T^{3} + 2561 T^{4} - 680 p T^{5} + 161 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 319 p T^{5} + 48 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 18 T + 83 T^{2} + 444 T^{3} - 6395 T^{4} + 444 p T^{5} + 83 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 335 p T^{5} + 67 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 - 3 T - 58 T^{2} + 375 T^{3} + 2761 T^{4} + 375 p T^{5} - 58 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T + 43 T^{2} + 74 T^{3} - 2745 T^{4} + 74 p T^{5} + 43 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 24 T + 233 T^{2} - 1860 T^{3} + 17101 T^{4} - 1860 p T^{5} + 233 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_4\times C_2$ | \( 1 - 5 T - 74 T^{2} - 25 T^{3} + 8391 T^{4} - 25 p T^{5} - 74 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 14 T + 223 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.25345529755595810795553480051, −6.81291544086306824660109300475, −6.60367905407952390735250794972, −6.58478828922934407134560457751, −6.34612848621877826433460219151, −6.03614606091503878477503009394, −6.00139876485733643193927234970, −5.97880275773719406422544194076, −5.23856603294510792078650407676, −5.23529251084677621597705920854, −4.94140487804094364451078787733, −4.75182229220907490860359202821, −4.46659492457912782593967049347, −4.41416873130107240549040119616, −4.06264824339039903348787447953, −3.81872281909296123270342472605, −3.25274122016133828112932054704, −2.95202200915168430424334925127, −2.65732911222119842214782010018, −2.61584728195844844788933366802, −2.49523049992971853739328317381, −1.69698751831322935612364222936, −1.65338320268291929804734447125, −1.54122079376950535838483140706, −1.07304978602893733409185747479,
1.07304978602893733409185747479, 1.54122079376950535838483140706, 1.65338320268291929804734447125, 1.69698751831322935612364222936, 2.49523049992971853739328317381, 2.61584728195844844788933366802, 2.65732911222119842214782010018, 2.95202200915168430424334925127, 3.25274122016133828112932054704, 3.81872281909296123270342472605, 4.06264824339039903348787447953, 4.41416873130107240549040119616, 4.46659492457912782593967049347, 4.75182229220907490860359202821, 4.94140487804094364451078787733, 5.23529251084677621597705920854, 5.23856603294510792078650407676, 5.97880275773719406422544194076, 6.00139876485733643193927234970, 6.03614606091503878477503009394, 6.34612848621877826433460219151, 6.58478828922934407134560457751, 6.60367905407952390735250794972, 6.81291544086306824660109300475, 7.25345529755595810795553480051
