| L(s) = 1 | + 2·4-s − 8·7-s − 4·13-s + 24·19-s − 2·25-s − 16·28-s − 18·31-s − 10·37-s + 34·49-s − 8·52-s − 10·61-s − 8·64-s + 18·67-s + 8·73-s + 48·76-s − 54·79-s + 32·91-s + 20·97-s − 4·100-s + 66·103-s − 10·109-s + 16·121-s − 36·124-s + 127-s + 131-s − 192·133-s + 137-s + ⋯ |
| L(s) = 1 | + 4-s − 3.02·7-s − 1.10·13-s + 5.50·19-s − 2/5·25-s − 3.02·28-s − 3.23·31-s − 1.64·37-s + 34/7·49-s − 1.10·52-s − 1.28·61-s − 64-s + 2.19·67-s + 0.936·73-s + 5.50·76-s − 6.07·79-s + 3.35·91-s + 2.03·97-s − 2/5·100-s + 6.50·103-s − 0.957·109-s + 1.45·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s − 16.6·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2600726467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2600726467\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 16 T^{2} + 135 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 32 T^{2} + 735 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 40 T^{2} - 609 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 8 T^{2} - 2745 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 64 T^{2} + 615 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 22 T^{2} - 7437 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
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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
−7.33954030476246801607655133222, −7.26850089908115807630898782317, −7.13107352604769912922872734538, −7.06627179263821952092969281116, −6.46470033675729656076424153023, −6.19474349965050336560396783214, −6.00150021174281633718225225959, −5.91756337016283453576551807467, −5.63561579474437703779566111735, −5.44116870148766207212109602560, −4.96138991969217475995384331888, −4.91082626868170050263950991477, −4.86591354183775118209374626594, −4.06357833971198367598073755247, −3.72389265616846728367821978826, −3.45246634867849467478988326928, −3.40481054725593714985678936477, −3.18413435071329298028850017681, −3.11574873380778136081104632195, −2.53702257141894537408450470620, −2.32111898726139404263118187507, −1.98313523846912663254530051732, −1.21518807804577845651613813185, −1.17345610309763894201329371499, −0.13679411927205685244502742793,
0.13679411927205685244502742793, 1.17345610309763894201329371499, 1.21518807804577845651613813185, 1.98313523846912663254530051732, 2.32111898726139404263118187507, 2.53702257141894537408450470620, 3.11574873380778136081104632195, 3.18413435071329298028850017681, 3.40481054725593714985678936477, 3.45246634867849467478988326928, 3.72389265616846728367821978826, 4.06357833971198367598073755247, 4.86591354183775118209374626594, 4.91082626868170050263950991477, 4.96138991969217475995384331888, 5.44116870148766207212109602560, 5.63561579474437703779566111735, 5.91756337016283453576551807467, 6.00150021174281633718225225959, 6.19474349965050336560396783214, 6.46470033675729656076424153023, 7.06627179263821952092969281116, 7.13107352604769912922872734538, 7.26850089908115807630898782317, 7.33954030476246801607655133222
