| L(s) = 1 | − 2-s + 3·5-s − 7-s + 3·9-s − 3·10-s − 11-s + 14-s − 17-s − 3·18-s − 19-s + 22-s + 6·25-s − 29-s + 34-s − 3·35-s + 38-s − 41-s − 43-s + 9·45-s − 6·50-s − 3·55-s + 58-s − 61-s − 3·63-s − 67-s + 3·70-s − 73-s + ⋯ |
| L(s) = 1 | − 2-s + 3·5-s − 7-s + 3·9-s − 3·10-s − 11-s + 14-s − 17-s − 3·18-s − 19-s + 22-s + 6·25-s − 29-s + 34-s − 3·35-s + 38-s − 41-s − 43-s + 9·45-s − 6·50-s − 3·55-s + 58-s − 61-s − 3·63-s − 67-s + 3·70-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5984734811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5984734811\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 107 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 67 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.906757694873072278689834095812, −9.527908122818056024124474424087, −9.463432795152910438958705270780, −9.241541822244537908891325200430, −8.795700638787193756020428334410, −8.540715632245741624162724795827, −8.388477665810043337945709438901, −7.67342018277742728715926375460, −7.34010203515956043348388656078, −7.00750874914935526178488014842, −6.87618428171147153180490134446, −6.51379586928985269625267836273, −6.31872134854547838858742969326, −5.83168506435481595520508771463, −5.65647829169602576387120462530, −5.17162941320833479001921604540, −4.71807635874933895606621314285, −4.48007720362656186316787000439, −4.26673127917785263132289690137, −3.28931895286286894011030684280, −3.23337952857169682004924550635, −2.26155132285538243848687768947, −2.25262003142672894754897117258, −1.60012331089677639396685172043, −1.39943739559820658234890411090,
1.39943739559820658234890411090, 1.60012331089677639396685172043, 2.25262003142672894754897117258, 2.26155132285538243848687768947, 3.23337952857169682004924550635, 3.28931895286286894011030684280, 4.26673127917785263132289690137, 4.48007720362656186316787000439, 4.71807635874933895606621314285, 5.17162941320833479001921604540, 5.65647829169602576387120462530, 5.83168506435481595520508771463, 6.31872134854547838858742969326, 6.51379586928985269625267836273, 6.87618428171147153180490134446, 7.00750874914935526178488014842, 7.34010203515956043348388656078, 7.67342018277742728715926375460, 8.388477665810043337945709438901, 8.540715632245741624162724795827, 8.795700638787193756020428334410, 9.241541822244537908891325200430, 9.463432795152910438958705270780, 9.527908122818056024124474424087, 9.906757694873072278689834095812
