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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.381121446\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.381121446\) |
\(L(1)\) |
\(\approx\) |
\(1.191621100\) |
\(L(1)\) |
\(\approx\) |
\(1.191621100\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 563 | \( 1 \) |
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 | \( 1 - T \) |
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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
−23.44398396056945250483541858602, −22.04688056142865267272281319893, −20.85236240398872433793148304114, −20.49367029895660009637199282704, −19.753442804007359156945374507126, −18.7179395741160821605018920845, −18.5356527640599964066818025206, −17.21224114116395438894820804909, −16.34269453977467928879784966058, −15.47757706224107021636988662965, −14.78364876645570558289215952034, −14.07998120294258143747645305513, −12.65908168737957057819821146728, −11.640812195304594641934990461882, −11.1015026366101615365813018130, −9.93214053653488081166099255384, −8.86344094072557433132551048165, −8.43850349534664769136122502358, −7.49723553068968270930933155930, −6.95543557580235650221803021336, −5.326258992365120469678467119688, −3.81549781393358963936410474048, −3.246419659457123454784826618553, −1.687605946571056968139985934980, −1.00820339867981708478766173874,
1.00820339867981708478766173874, 1.687605946571056968139985934980, 3.246419659457123454784826618553, 3.81549781393358963936410474048, 5.326258992365120469678467119688, 6.95543557580235650221803021336, 7.49723553068968270930933155930, 8.43850349534664769136122502358, 8.86344094072557433132551048165, 9.93214053653488081166099255384, 11.1015026366101615365813018130, 11.640812195304594641934990461882, 12.65908168737957057819821146728, 14.07998120294258143747645305513, 14.78364876645570558289215952034, 15.47757706224107021636988662965, 16.34269453977467928879784966058, 17.21224114116395438894820804909, 18.5356527640599964066818025206, 18.7179395741160821605018920845, 19.753442804007359156945374507126, 20.49367029895660009637199282704, 20.85236240398872433793148304114, 22.04688056142865267272281319893, 23.44398396056945250483541858602