| L(s) = 1 | − 2-s − 3-s + 5·5-s + 6-s − 7-s − 5·10-s − 13-s + 14-s − 5·15-s − 19-s + 21-s − 23-s + 15·25-s + 26-s + 5·30-s − 31-s − 5·35-s + 10·37-s + 38-s + 39-s − 42-s + 46-s − 47-s − 15·50-s + 57-s + 62-s − 5·65-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 5·5-s + 6-s − 7-s − 5·10-s − 13-s + 14-s − 5·15-s − 19-s + 21-s − 23-s + 15·25-s + 26-s + 5·30-s − 31-s − 5·35-s + 10·37-s + 38-s + 39-s − 42-s + 46-s − 47-s − 15·50-s + 57-s + 62-s − 5·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 331^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 331^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.079508444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079508444\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{5} \) |
| 331 | $C_1$ | \( ( 1 - T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 13 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 19 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 23 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 37 | $C_1$ | \( ( 1 - T )^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 71 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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
−5.98135163032228775362872207075, −5.89846072971825239522757224990, −5.65509050915975873571274619548, −5.61038178462835411548492163550, −5.21579508019638567134815285858, −4.98348881849903284444061852779, −4.75120357739874623955362510926, −4.69597826784836267446852068196, −4.58481875475579514586569004199, −4.35561221621355356276414423314, −4.26851437861063143179338033019, −3.91757166777038406649130491305, −3.42799586963469749358947380624, −3.24888113531475598061820609473, −2.90349290197233869211897064871, −2.82253364966760575051771541171, −2.76436624852560821098679879824, −2.40639542340452916945388013414, −2.21796229737834007761111342480, −2.11269550650567183783705143679, −2.00394869593543275099363380243, −1.64731637203887670119570266464, −1.09819984465470563206824184079, −0.993444736305504792211507648180, −0.830369497174090328070813383494,
0.830369497174090328070813383494, 0.993444736305504792211507648180, 1.09819984465470563206824184079, 1.64731637203887670119570266464, 2.00394869593543275099363380243, 2.11269550650567183783705143679, 2.21796229737834007761111342480, 2.40639542340452916945388013414, 2.76436624852560821098679879824, 2.82253364966760575051771541171, 2.90349290197233869211897064871, 3.24888113531475598061820609473, 3.42799586963469749358947380624, 3.91757166777038406649130491305, 4.26851437861063143179338033019, 4.35561221621355356276414423314, 4.58481875475579514586569004199, 4.69597826784836267446852068196, 4.75120357739874623955362510926, 4.98348881849903284444061852779, 5.21579508019638567134815285858, 5.61038178462835411548492163550, 5.65509050915975873571274619548, 5.89846072971825239522757224990, 5.98135163032228775362872207075
