| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 5·16-s + 2·19-s − 6·20-s + 2·23-s + 3·25-s + 6·32-s + 4·38-s − 8·40-s + 4·46-s − 2·47-s + 49-s + 6·50-s − 2·53-s + 7·64-s + 6·76-s − 10·80-s + 6·92-s − 4·94-s − 4·95-s + 2·98-s + 9·100-s − 4·106-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 5·16-s + 2·19-s − 6·20-s + 2·23-s + 3·25-s + 6·32-s + 4·38-s − 8·40-s + 4·46-s − 2·47-s + 49-s + 6·50-s − 2·53-s + 7·64-s + 6·76-s − 10·80-s + 6·92-s − 4·94-s − 4·95-s + 2·98-s + 9·100-s − 4·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.665112702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.665112702\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.843102154575180603226795560812, −8.487711935563782900956867445667, −7.88339696295926010696013236388, −7.84008506744088215445243132544, −7.31857335548969746701501841195, −7.18956619119126998748786641788, −6.78212316264563716005023414725, −6.39318934248509787989908275759, −5.99070392701907244620605910055, −5.22944657780010103416467180690, −5.07087499591302126826955994498, −4.91126924562706515062415850639, −4.38526473926340885184688233917, −3.94683578935309370283610330675, −3.38558437304163758272421347164, −3.35776322423203467084332004954, −2.89409784019407831624210942256, −2.47743341531067624936286404992, −1.37798830320113286732611904974, −1.12122624679955138862709153639,
1.12122624679955138862709153639, 1.37798830320113286732611904974, 2.47743341531067624936286404992, 2.89409784019407831624210942256, 3.35776322423203467084332004954, 3.38558437304163758272421347164, 3.94683578935309370283610330675, 4.38526473926340885184688233917, 4.91126924562706515062415850639, 5.07087499591302126826955994498, 5.22944657780010103416467180690, 5.99070392701907244620605910055, 6.39318934248509787989908275759, 6.78212316264563716005023414725, 7.18956619119126998748786641788, 7.31857335548969746701501841195, 7.84008506744088215445243132544, 7.88339696295926010696013236388, 8.487711935563782900956867445667, 8.843102154575180603226795560812
