| L(s) = 1 | − 2-s + 5·9-s − 13-s − 5·18-s − 23-s + 5·25-s + 26-s − 41-s + 46-s + 5·49-s − 5·50-s − 61-s − 67-s − 79-s + 15·81-s + 82-s − 83-s − 89-s − 5·98-s − 103-s − 109-s − 113-s − 5·117-s + 5·121-s + 122-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2-s + 5·9-s − 13-s − 5·18-s − 23-s + 5·25-s + 26-s − 41-s + 46-s + 5·49-s − 5·50-s − 61-s − 67-s − 79-s + 15·81-s + 82-s − 83-s − 89-s − 5·98-s − 103-s − 109-s − 113-s − 5·117-s + 5·121-s + 122-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1303^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1303^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9750123453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750123453\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1303 | $C_1$ | \( ( 1 - T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_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_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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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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
−6.15906246070400453764973182937, −5.95610174947768459102177880246, −5.59781757676187892478949981359, −5.38764584205613917884399296923, −5.38701991796606133332280685529, −4.86799811902599925316447841510, −4.77573403747939430720728191678, −4.68728365651808918880788767128, −4.61905077612086876373932789728, −4.60827826990408734939009745928, −4.04644317575707131527640531549, −3.95552809636614571698924215850, −3.91722517471613998744552799378, −3.81750756496842638922539402069, −3.33890072597395085145558201184, −3.06814001176919337519022237188, −2.89525105345502872843806324297, −2.65740367150367347755910876893, −2.20984866761775695625174137821, −2.18553531301260884791012770538, −1.97461115041838847854311288998, −1.41163479920336533613124313004, −1.11150612389930148990524845605, −1.09945564540253261297300848596, −1.08036710951809838189526682606,
1.08036710951809838189526682606, 1.09945564540253261297300848596, 1.11150612389930148990524845605, 1.41163479920336533613124313004, 1.97461115041838847854311288998, 2.18553531301260884791012770538, 2.20984866761775695625174137821, 2.65740367150367347755910876893, 2.89525105345502872843806324297, 3.06814001176919337519022237188, 3.33890072597395085145558201184, 3.81750756496842638922539402069, 3.91722517471613998744552799378, 3.95552809636614571698924215850, 4.04644317575707131527640531549, 4.60827826990408734939009745928, 4.61905077612086876373932789728, 4.68728365651808918880788767128, 4.77573403747939430720728191678, 4.86799811902599925316447841510, 5.38701991796606133332280685529, 5.38764584205613917884399296923, 5.59781757676187892478949981359, 5.95610174947768459102177880246, 6.15906246070400453764973182937
