| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s − 21-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 29-s − 30-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s − 21-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 29-s − 30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9915770035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9915770035\) |
| \(L(1)\) |
\(\approx\) |
\(0.9472258250\) |
| \(L(1)\) |
\(\approx\) |
\(0.9472258250\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−45.03443041544910895447298842202, −44.22866228755402915071954091702, −42.63941507630696800457616302101, −41.49315644722970854417895464060, −38.95426138432898441184537352440, −37.66796277038011510457643735308, −36.65347629447046358323959856101, −35.485085187617277263778295767982, −33.52280264533323492464912935790, −32.10991724081501633365902470335, −29.97469117254131187808345346222, −28.832343001278275591309074781924, −26.78689568412247833195118491511, −25.68596243541730222680937264738, −24.67283686109554841773882275166, −21.638177818276573522470778958792, −20.06759332864609363804035036489, −18.79724653616265280786110493309, −16.99071070103014279771587435499, −15.10915824669017999758236697486, −13.04011532881724693326998338500, −10.10833735739279668002774158265, −8.97128436849938383436174724130, −6.80070840838651795033630280388, −2.47724371122923425905188980494,
2.47724371122923425905188980494, 6.80070840838651795033630280388, 8.97128436849938383436174724130, 10.10833735739279668002774158265, 13.04011532881724693326998338500, 15.10915824669017999758236697486, 16.99071070103014279771587435499, 18.79724653616265280786110493309, 20.06759332864609363804035036489, 21.638177818276573522470778958792, 24.67283686109554841773882275166, 25.68596243541730222680937264738, 26.78689568412247833195118491511, 28.832343001278275591309074781924, 29.97469117254131187808345346222, 32.10991724081501633365902470335, 33.52280264533323492464912935790, 35.485085187617277263778295767982, 36.65347629447046358323959856101, 37.66796277038011510457643735308, 38.95426138432898441184537352440, 41.49315644722970854417895464060, 42.63941507630696800457616302101, 44.22866228755402915071954091702, 45.03443041544910895447298842202
