| L(s) = 1 | + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 15·12-s − 13-s + 15-s + 70·16-s − 17-s − 15·20-s − 23-s − 35·24-s − 5·26-s + 5·30-s − 31-s + 126·32-s − 5·34-s + 39-s − 35·40-s − 41-s − 43-s − 5·46-s − 47-s − 70·48-s + ⋯ |
| L(s) = 1 | + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 15·12-s − 13-s + 15-s + 70·16-s − 17-s − 15·20-s − 23-s − 35·24-s − 5·26-s + 5·30-s − 31-s + 126·32-s − 5·34-s + 39-s − 35·40-s − 41-s − 43-s − 5·46-s − 47-s − 70·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 389^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 389^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(22.19985914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.19985914\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{5} \) |
| 389 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 3 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 5 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 23 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 41 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 43 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.70647687957340990039979100266, −5.52997188028739674836341899753, −5.48288231280381090661027491858, −5.47474108708325571559315914775, −5.02361144378662908833203211806, −5.01491994217826736217162124336, −4.85296808825697812866993371759, −4.62188794618259477461587979248, −4.49054114558404792760852536430, −4.17493239508791969396279437183, −4.06088443410740939564103080535, −4.00610966058322910537090848684, −3.84835271805551361900805583785, −3.64319630873735181423597846675, −3.49912874864971178077019519706, −3.12492100480765273131802162686, −2.95503899065309717953781813472, −2.72833231484897467628260317433, −2.55647392491648338285116298367, −2.51461661123882110754334543961, −1.98018778006844303583063626452, −1.87667527167217317290083399540, −1.85741019919461884781263636767, −1.31320331233899412755551157708, −1.03561789011630349525983905368,
1.03561789011630349525983905368, 1.31320331233899412755551157708, 1.85741019919461884781263636767, 1.87667527167217317290083399540, 1.98018778006844303583063626452, 2.51461661123882110754334543961, 2.55647392491648338285116298367, 2.72833231484897467628260317433, 2.95503899065309717953781813472, 3.12492100480765273131802162686, 3.49912874864971178077019519706, 3.64319630873735181423597846675, 3.84835271805551361900805583785, 4.00610966058322910537090848684, 4.06088443410740939564103080535, 4.17493239508791969396279437183, 4.49054114558404792760852536430, 4.62188794618259477461587979248, 4.85296808825697812866993371759, 5.01491994217826736217162124336, 5.02361144378662908833203211806, 5.47474108708325571559315914775, 5.48288231280381090661027491858, 5.52997188028739674836341899753, 5.70647687957340990039979100266
