| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3583 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3583 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.418612901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.418612901\) |
| \(L(1)\) |
\(\approx\) |
\(1.522034394\) |
| \(L(1)\) |
\(\approx\) |
\(1.522034394\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3583 | \( 1 \) |
| 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
−18.64461311393192855087628690923, −17.54583087267053589195963139971, −17.06757017980170731300190014044, −16.40063845032062464751736060497, −15.7816814325486296594010301652, −14.90096514990298433590383101635, −14.57346761998341866890850063386, −13.83782194462815341395119720319, −12.52268556683595038741515365357, −12.29472554101801272676248336061, −11.78419662501509704926752825738, −11.09489413445673271261470457707, −10.55951715289165277173328201017, −9.672877053668430378838490429331, −8.333568266545099038588293845956, −7.71885987870912022267886030226, −6.97819773229894674293819990010, −6.41723458291287449034169113582, −5.35715998001714510944306307972, −4.95031752714679144310681661270, −4.05115467542618849210695658509, −3.779487547709490748728448465324, −2.37696052998508303105279439381, −1.52350999498830151602668139211, −0.62883778355454771969324264342,
0.62883778355454771969324264342, 1.52350999498830151602668139211, 2.37696052998508303105279439381, 3.779487547709490748728448465324, 4.05115467542618849210695658509, 4.95031752714679144310681661270, 5.35715998001714510944306307972, 6.41723458291287449034169113582, 6.97819773229894674293819990010, 7.71885987870912022267886030226, 8.333568266545099038588293845956, 9.672877053668430378838490429331, 10.55951715289165277173328201017, 11.09489413445673271261470457707, 11.78419662501509704926752825738, 12.29472554101801272676248336061, 12.52268556683595038741515365357, 13.83782194462815341395119720319, 14.57346761998341866890850063386, 14.90096514990298433590383101635, 15.7816814325486296594010301652, 16.40063845032062464751736060497, 17.06757017980170731300190014044, 17.54583087267053589195963139971, 18.64461311393192855087628690923
