| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 12·7-s + 5·8-s − 2·9-s + 3·11-s + 2·12-s + 6·13-s + 24·14-s + 5·16-s + 7·17-s − 4·18-s + 5·19-s + 12·21-s + 6·22-s + 6·23-s + 5·24-s + 12·26-s + 24·28-s − 15·29-s − 12·31-s − 2·32-s + 3·33-s + 14·34-s − 4·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 4.53·7-s + 1.76·8-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 1.66·13-s + 6.41·14-s + 5/4·16-s + 1.69·17-s − 0.942·18-s + 1.14·19-s + 2.61·21-s + 1.27·22-s + 1.25·23-s + 1.02·24-s + 2.35·26-s + 4.53·28-s − 2.78·29-s − 2.15·31-s − 0.353·32-s + 0.522·33-s + 2.40·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.71932009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.71932009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 - T + p T^{2} - 5 T^{3} + 16 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - 3 T - 2 T^{2} + 39 T^{3} - 95 T^{4} + 39 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 7 T + 2 T^{2} + 65 T^{3} - 169 T^{4} + 65 p T^{5} + 2 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 - 6 T - 7 T^{2} + 30 T^{3} + 361 T^{4} + 30 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + 15 T + 106 T^{2} + 675 T^{3} + 4171 T^{4} + 675 p T^{5} + 106 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 63 T^{2} + 14 p T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + 63 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 18 T + 107 T^{2} + 210 T^{3} + T^{4} + 210 p T^{5} + 107 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 3 T - 32 T^{2} + 219 T^{3} + 655 T^{4} + 219 p T^{5} - 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 18 T + 137 T^{2} + 990 T^{3} + 7951 T^{4} + 990 p T^{5} + 137 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T - 37 T^{2} + 360 T^{3} + 121 T^{4} + 360 p T^{5} - 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 15 T + 31 T^{2} - 15 p T^{3} - 10424 T^{4} - 15 p^{2} T^{5} + 31 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 + T - 29 T^{2} + p T^{3} + p^{2} T^{4} )( 1 + 11 T + 51 T^{2} + 11 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^2:C_4$ | \( 1 + 18 T + 77 T^{2} - 1110 T^{3} - 17669 T^{4} - 1110 p T^{5} + 77 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 9 T - 37 T^{2} - 165 T^{3} + 4096 T^{4} - 165 p T^{5} - 37 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 715 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 6 T - 7 T^{2} - 720 T^{3} + 11221 T^{4} - 720 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 30 T + 451 T^{2} + 5400 T^{3} + 56341 T^{4} + 5400 p T^{5} + 451 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 12 T - 43 T^{2} + 900 T^{3} - 1859 T^{4} + 900 p T^{5} - 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.64357488615353333304081278006, −7.54874302374735635294936186923, −7.19858437646776198408952664174, −7.17774897889637640638201346750, −6.99743636863288176826705985370, −6.04205954952820237389470179231, −5.95721728587460759671213193035, −5.91446712471854276113862244309, −5.55106444768791377223910327914, −5.54181899346438456754172192226, −4.98054999661851797107399223761, −4.81688616114965219914484008719, −4.74707649054486146018078675739, −4.53966820073703962365294411420, −4.41367792985774835793984553484, −3.66302559260902772792797634801, −3.62986753811717619225181640535, −3.48743510272140338323432857979, −3.34258169756407973780063863353, −2.42696057016447077656324968017, −2.33453867960443266575672794357, −1.63321595585907993392196370159, −1.62667811992059918095319748907, −1.35467041291886801359594419385, −1.26929525429130014977627004030,
1.26929525429130014977627004030, 1.35467041291886801359594419385, 1.62667811992059918095319748907, 1.63321595585907993392196370159, 2.33453867960443266575672794357, 2.42696057016447077656324968017, 3.34258169756407973780063863353, 3.48743510272140338323432857979, 3.62986753811717619225181640535, 3.66302559260902772792797634801, 4.41367792985774835793984553484, 4.53966820073703962365294411420, 4.74707649054486146018078675739, 4.81688616114965219914484008719, 4.98054999661851797107399223761, 5.54181899346438456754172192226, 5.55106444768791377223910327914, 5.91446712471854276113862244309, 5.95721728587460759671213193035, 6.04205954952820237389470179231, 6.99743636863288176826705985370, 7.17774897889637640638201346750, 7.19858437646776198408952664174, 7.54874302374735635294936186923, 7.64357488615353333304081278006
