L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 13-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 29-s − 31-s + 32-s − 34-s − 37-s − 38-s + 41-s − 43-s − 44-s + 46-s − 47-s + 52-s + 53-s − 58-s + 59-s − 61-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 13-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 29-s − 31-s + 32-s − 34-s − 37-s − 38-s + 41-s − 43-s − 44-s + 46-s − 47-s + 52-s + 53-s − 58-s + 59-s − 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.777644139\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.777644139\) |
\(L(1)\) |
\(\approx\) |
\(1.720148986\) |
\(L(1)\) |
\(\approx\) |
\(1.720148986\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 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 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.81183932642223432305890548289, −28.91694642959154523629622140391, −27.93650268603585835006525084610, −26.32200911279995332683334128787, −25.509932190826191270454116721049, −24.31915969183197091773299882641, −23.448412084519404177850939942, −22.589851476453794840498219310699, −21.34257419248581940898644944004, −20.68405044142393211855232045771, −19.47034002879828613436007948109, −18.20461932859221706711004561700, −16.73636122039523014164316730611, −15.67012050348698231277615318673, −14.827120768810874534393445019306, −13.39081440029833406196735481678, −12.862345237901543307432137965867, −11.29907981354199366805953260575, −10.555559845234520329443114663035, −8.68713481050641910740736755603, −7.26120229959507451601179000344, −6.04532098532612637992106866437, −4.83316627385172929886241582021, −3.49996851859653038990910140652, −2.04884908157592245734866028139,
2.04884908157592245734866028139, 3.49996851859653038990910140652, 4.83316627385172929886241582021, 6.04532098532612637992106866437, 7.26120229959507451601179000344, 8.68713481050641910740736755603, 10.555559845234520329443114663035, 11.29907981354199366805953260575, 12.862345237901543307432137965867, 13.39081440029833406196735481678, 14.827120768810874534393445019306, 15.67012050348698231277615318673, 16.73636122039523014164316730611, 18.20461932859221706711004561700, 19.47034002879828613436007948109, 20.68405044142393211855232045771, 21.34257419248581940898644944004, 22.589851476453794840498219310699, 23.448412084519404177850939942, 24.31915969183197091773299882641, 25.509932190826191270454116721049, 26.32200911279995332683334128787, 27.93650268603585835006525084610, 28.91694642959154523629622140391, 29.81183932642223432305890548289