L(s) = 1 | + 2-s − 3-s − 6-s + 11-s − 5·13-s − 17-s + 22-s + 5·25-s − 5·26-s − 29-s + 31-s − 33-s − 34-s + 37-s + 5·39-s + 47-s + 5·49-s + 5·50-s + 51-s − 53-s − 58-s − 5·59-s + 62-s − 66-s + 67-s + 73-s + 74-s + ⋯ |
L(s) = 1 | + 2-s − 3-s − 6-s + 11-s − 5·13-s − 17-s + 22-s + 5·25-s − 5·26-s − 29-s + 31-s − 33-s − 34-s + 37-s + 5·39-s + 47-s + 5·49-s + 5·50-s + 51-s − 53-s − 58-s − 5·59-s + 62-s − 66-s + 67-s + 73-s + 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 59^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 59^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5705881554\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5705881554\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 13 | $C_1$ | \( ( 1 + T )^{5} \) |
| 59 | $C_1$ | \( ( 1 + T )^{5} \) |
good | 2 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 11 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 17 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 29 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 31 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 37 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 53 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 73 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 79 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 83 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 89 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 97 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66701328319986792913968538831, −6.35519728035678787778307830872, −6.11590387023114144310481352429, −5.83853721034215461320604547788, −5.68845938328872724006946840069, −5.43941577739673935666860719115, −5.24710133049174967645768707191, −5.01537309415466201455841449765, −5.00634977060005652825482672749, −4.70025422853543237855090900212, −4.49841166915600049258646154768, −4.49816692483038303996229619504, −4.48319346621222367677250540756, −4.07604152236618056629941811918, −3.89020719379757135349492093278, −3.28097313904114313198704208218, −3.27548626415212907585360345153, −2.87747797864626746804129423045, −2.76242806472573669333939517736, −2.50393477711347000546588339533, −2.27510180388573178701149035976, −2.21429517618482507700647045541, −1.63861239504145141470361874049, −1.05861150814027667836860376426, −0.74043603966894655226262109939,
0.74043603966894655226262109939, 1.05861150814027667836860376426, 1.63861239504145141470361874049, 2.21429517618482507700647045541, 2.27510180388573178701149035976, 2.50393477711347000546588339533, 2.76242806472573669333939517736, 2.87747797864626746804129423045, 3.27548626415212907585360345153, 3.28097313904114313198704208218, 3.89020719379757135349492093278, 4.07604152236618056629941811918, 4.48319346621222367677250540756, 4.49816692483038303996229619504, 4.49841166915600049258646154768, 4.70025422853543237855090900212, 5.00634977060005652825482672749, 5.01537309415466201455841449765, 5.24710133049174967645768707191, 5.43941577739673935666860719115, 5.68845938328872724006946840069, 5.83853721034215461320604547788, 6.11590387023114144310481352429, 6.35519728035678787778307830872, 6.66701328319986792913968538831