| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.271084717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271084717\) |
| \(L(1)\) |
\(\approx\) |
\(1.051432743\) |
| \(L(1)\) |
\(\approx\) |
\(1.051432743\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 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
−25.03312323355877569256927246429, −24.571465665094750959944301944663, −23.19807395710073714044914858749, −21.75444457292945269268580271162, −21.1363913363952724941812258936, −20.28732608097418441776876219459, −19.42003310837954826012405303358, −18.70448508358903174472893551615, −17.9180046203487492466139983205, −16.80633840124229383463363203496, −16.0092635316319985724306018352, −15.08088093855244640117076402841, −14.07459379900878421651213876088, −13.05139191461974555930055674087, −12.26141795566903082916933547516, −10.57165516284579865992476526849, −9.693299789912410096177173196679, −9.54008036334649211408681853221, −8.19882232111887051563226637720, −7.33922369648524434989958828981, −6.37496497907013163710196605625, −5.09643259621438734752154437742, −2.95187641263826426984315717468, −2.736598166556370856746387524665, −1.22568669313072189808564974638,
1.22568669313072189808564974638, 2.736598166556370856746387524665, 2.95187641263826426984315717468, 5.09643259621438734752154437742, 6.37496497907013163710196605625, 7.33922369648524434989958828981, 8.19882232111887051563226637720, 9.54008036334649211408681853221, 9.693299789912410096177173196679, 10.57165516284579865992476526849, 12.26141795566903082916933547516, 13.05139191461974555930055674087, 14.07459379900878421651213876088, 15.08088093855244640117076402841, 16.0092635316319985724306018352, 16.80633840124229383463363203496, 17.9180046203487492466139983205, 18.70448508358903174472893551615, 19.42003310837954826012405303358, 20.28732608097418441776876219459, 21.1363913363952724941812258936, 21.75444457292945269268580271162, 23.19807395710073714044914858749, 24.571465665094750959944301944663, 25.03312323355877569256927246429
