| L(s) = 1 | − 2-s − 5-s − 7-s + 5·9-s + 10-s − 11-s + 5·13-s + 14-s − 17-s − 5·18-s + 22-s − 23-s − 5·26-s − 29-s − 31-s + 34-s + 35-s − 41-s − 5·45-s + 46-s + 55-s + 58-s + 62-s − 5·63-s − 5·65-s + 5·67-s − 70-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s − 7-s + 5·9-s + 10-s − 11-s + 5·13-s + 14-s − 17-s − 5·18-s + 22-s − 23-s − 5·26-s − 29-s − 31-s + 34-s + 35-s − 41-s − 5·45-s + 46-s + 55-s + 58-s + 62-s − 5·63-s − 5·65-s + 5·67-s − 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5050803489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5050803489\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{5} \) |
| 67 | $C_1$ | \( ( 1 - T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 11 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
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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
−6.41521777442232251601523839744, −6.38007336214568427242312476651, −6.22579924089976122148137893873, −5.96606895551648770807377357818, −5.67699795498372690253040182018, −5.51700652968924302243666686000, −5.06816699709763172518015467504, −5.02011044839378537216726805350, −4.99486295131559950778141150225, −4.49517242692864748899786132279, −4.12663462674993278134103642721, −4.06186531403159952222565163371, −4.02487493966216550249959192141, −3.93534756426053421465410465936, −3.65639697030646127513698771091, −3.55612899544479759184163460845, −3.41765566975277048208790339315, −3.10648689350912656989694630407, −2.45630345460003225869083086521, −2.21769598878851124827386249683, −2.00370362099285550622550028097, −1.55079578150419719949774376287, −1.48755455653452536025097078877, −1.16451097523120678313979026605, −0.913512058416947344120449872157,
0.913512058416947344120449872157, 1.16451097523120678313979026605, 1.48755455653452536025097078877, 1.55079578150419719949774376287, 2.00370362099285550622550028097, 2.21769598878851124827386249683, 2.45630345460003225869083086521, 3.10648689350912656989694630407, 3.41765566975277048208790339315, 3.55612899544479759184163460845, 3.65639697030646127513698771091, 3.93534756426053421465410465936, 4.02487493966216550249959192141, 4.06186531403159952222565163371, 4.12663462674993278134103642721, 4.49517242692864748899786132279, 4.99486295131559950778141150225, 5.02011044839378537216726805350, 5.06816699709763172518015467504, 5.51700652968924302243666686000, 5.67699795498372690253040182018, 5.96606895551648770807377357818, 6.22579924089976122148137893873, 6.38007336214568427242312476651, 6.41521777442232251601523839744
