| L(s) = 1 | − 2·3-s − 2·5-s + 2·9-s − 4·11-s + 12·13-s + 4·15-s + 2·19-s + 8·23-s + 6·25-s + 8·27-s + 4·31-s + 8·33-s − 24·39-s − 16·41-s − 8·43-s − 4·45-s + 12·47-s + 20·53-s + 8·55-s − 4·57-s − 14·59-s − 18·61-s − 24·65-s + 8·67-s − 16·69-s − 16·71-s − 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 2/3·9-s − 1.20·11-s + 3.32·13-s + 1.03·15-s + 0.458·19-s + 1.66·23-s + 6/5·25-s + 1.53·27-s + 0.718·31-s + 1.39·33-s − 3.84·39-s − 2.49·41-s − 1.21·43-s − 0.596·45-s + 1.75·47-s + 2.74·53-s + 1.07·55-s − 0.529·57-s − 1.82·59-s − 2.30·61-s − 2.97·65-s + 0.977·67-s − 1.92·69-s − 1.89·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752374701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752374701\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 8 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 64 T^{3} - 261 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 8 p T^{5} - 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 - 4 T - 30 T^{2} + 64 T^{3} + 659 T^{4} + 64 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 54 T^{2} + 1547 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 34 T^{2} - 192 T^{3} + 3123 T^{4} - 192 p T^{5} + 34 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 34 T^{2} + 616 T^{3} + 11199 T^{4} + 616 p T^{5} + 34 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 126 T^{2} + 1368 T^{3} + 15719 T^{4} + 1368 p T^{5} + 126 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 42 T^{2} - 528 T^{3} - 5437 T^{4} - 528 p T^{5} + 42 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 30 T^{2} + 512 T^{3} - 2461 T^{4} + 512 p T^{5} - 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.73720565302558575970059812967, −6.56227331910042142521843247123, −6.18128917098825135404061259194, −6.12472487427789030911784089386, −5.96750365124654016928026386027, −5.63041276370892819913542852562, −5.41526400785759144823056207362, −5.08219931586009969905089441816, −5.07844481449070923253486601402, −4.78109764087852007924899706430, −4.55048784368215754485650245562, −4.50382408079034950236237395687, −3.88617041770536397016379778328, −3.72614115959193451790583610042, −3.69524108036922895889531083965, −3.30466200013414942013579493394, −3.01842157415827321798706251594, −2.95594503479187014769687195274, −2.57289404709095337168354751736, −2.24093177878760693690444564115, −1.56219184122776208435135310645, −1.42797715517561468564726213099, −1.05055328484236921286797391795, −0.877774754367569730397390776536, −0.34602097072526359280066322291,
0.34602097072526359280066322291, 0.877774754367569730397390776536, 1.05055328484236921286797391795, 1.42797715517561468564726213099, 1.56219184122776208435135310645, 2.24093177878760693690444564115, 2.57289404709095337168354751736, 2.95594503479187014769687195274, 3.01842157415827321798706251594, 3.30466200013414942013579493394, 3.69524108036922895889531083965, 3.72614115959193451790583610042, 3.88617041770536397016379778328, 4.50382408079034950236237395687, 4.55048784368215754485650245562, 4.78109764087852007924899706430, 5.07844481449070923253486601402, 5.08219931586009969905089441816, 5.41526400785759144823056207362, 5.63041276370892819913542852562, 5.96750365124654016928026386027, 6.12472487427789030911784089386, 6.18128917098825135404061259194, 6.56227331910042142521843247123, 6.73720565302558575970059812967
