| L(s) = 1 | + 2-s + 2·3-s + 5-s + 2·6-s − 7-s + 3·9-s + 10-s + 11-s + 8·13-s − 14-s + 2·15-s − 4·17-s + 3·18-s + 3·19-s − 2·21-s + 22-s + 16·23-s + 8·26-s + 10·27-s + 10·29-s + 2·30-s − 8·31-s − 32-s + 2·33-s − 4·34-s − 35-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 9-s + 0.316·10-s + 0.301·11-s + 2.21·13-s − 0.267·14-s + 0.516·15-s − 0.970·17-s + 0.707·18-s + 0.688·19-s − 0.436·21-s + 0.213·22-s + 3.33·23-s + 1.56·26-s + 1.92·27-s + 1.85·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s + 0.348·33-s − 0.685·34-s − 0.169·35-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.35253596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.35253596\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + 19 T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 8 T + 11 T^{2} + 86 T^{3} - 491 T^{4} + 86 p T^{5} + 11 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 4 T - 11 T^{2} - 52 T^{3} + 69 T^{4} - 52 p T^{5} - 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 3 T + 35 T^{2} - 63 T^{3} + 784 T^{4} - 63 p T^{5} + 35 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 200 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} )( 1 + 19 T + 151 T^{2} + 19 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2:C_4$ | \( 1 - 4 T + 59 T^{2} - 398 T^{3} + 1669 T^{4} - 398 p T^{5} + 59 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 8 T - 7 T^{2} - 36 T^{3} + 1925 T^{4} - 36 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 12 T + 17 T^{2} - 570 T^{3} - 5819 T^{4} - 570 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 13 T^{2} + 330 T^{3} + 2149 T^{4} + 330 p T^{5} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 + 22 T + 123 T^{2} - 886 T^{3} - 15295 T^{4} - 886 p T^{5} + 123 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 + 22 T + 173 T^{2} + 1084 T^{3} + 9965 T^{4} + 1084 p T^{5} + 173 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 8 T - 39 T^{2} - 4 T^{3} + 5669 T^{4} - 4 p T^{5} - 39 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 4 T - 63 T^{2} - 568 T^{3} + 2705 T^{4} - 568 p T^{5} - 63 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 13 T - 19 T^{2} + 951 T^{3} - 6196 T^{4} + 951 p T^{5} - 19 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 23 T + 299 T^{2} + 23 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 7 T - 3 T^{2} + 775 T^{3} - 4204 T^{4} + 775 p T^{5} - 3 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.40440937347368519609584993043, −7.21871121269412303424888371351, −6.99082132677765811607329851892, −6.57522159051783907433885961997, −6.44499245866263455244033299827, −6.32830580614028160642863279334, −6.05400179081440779123184935907, −5.76091704853597887126258702170, −5.71602392363153508871007039461, −5.07642476909370383560368844732, −4.98635701782804356269810427486, −4.69320890665250921823069330744, −4.46538170804989560808534817160, −4.30849656905041480001551140722, −4.09173887897287689054320628642, −3.60241641422162619131473061889, −3.31163370614067613175126902385, −3.13703218472527921766396914263, −2.94760118567541109248047283790, −2.64709525694873745018113978075, −2.53523191131364105535663944881, −1.60935992578123152568348996046, −1.42205800234605816658568412209, −1.33616528836126960231519522221, −0.71150018506487859814641346662,
0.71150018506487859814641346662, 1.33616528836126960231519522221, 1.42205800234605816658568412209, 1.60935992578123152568348996046, 2.53523191131364105535663944881, 2.64709525694873745018113978075, 2.94760118567541109248047283790, 3.13703218472527921766396914263, 3.31163370614067613175126902385, 3.60241641422162619131473061889, 4.09173887897287689054320628642, 4.30849656905041480001551140722, 4.46538170804989560808534817160, 4.69320890665250921823069330744, 4.98635701782804356269810427486, 5.07642476909370383560368844732, 5.71602392363153508871007039461, 5.76091704853597887126258702170, 6.05400179081440779123184935907, 6.32830580614028160642863279334, 6.44499245866263455244033299827, 6.57522159051783907433885961997, 6.99082132677765811607329851892, 7.21871121269412303424888371351, 7.40440937347368519609584993043
