| L(s) = 1 | + 2-s − 5-s − 7-s − 8-s − 10-s + 2·13-s − 14-s − 16-s + 2·19-s + 25-s + 2·26-s + 29-s + 35-s + 2·38-s + 40-s − 43-s + 50-s + 53-s + 56-s + 58-s + 59-s + 64-s − 2·65-s + 67-s + 70-s − 2·71-s + 80-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 7-s − 8-s − 10-s + 2·13-s − 14-s − 16-s + 2·19-s + 25-s + 2·26-s + 29-s + 35-s + 2·38-s + 40-s − 43-s + 50-s + 53-s + 56-s + 58-s + 59-s + 64-s − 2·65-s + 67-s + 70-s − 2·71-s + 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.440093309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440093309\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.774975206452880198988454127607, −8.275493696624421167515679595203, −7.20797069106514721874238687237, −6.48726052312562094555691538730, −5.79553068964746398695460871450, −4.99277492651023153742512674474, −4.02045941682050681089505072235, −3.44169671887910210063988449314, −2.98991309510037480883188611876, −0.963205255221829721369699682375,
0.963205255221829721369699682375, 2.98991309510037480883188611876, 3.44169671887910210063988449314, 4.02045941682050681089505072235, 4.99277492651023153742512674474, 5.79553068964746398695460871450, 6.48726052312562094555691538730, 7.20797069106514721874238687237, 8.275493696624421167515679595203, 8.774975206452880198988454127607
