| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.994228569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994228569\) |
| \(L(1)\) |
\(\approx\) |
\(2.418356383\) |
| \(L(1)\) |
\(\approx\) |
\(2.418356383\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 \) |
| 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
−25.93121454419649980103844702736, −25.23900430848409851967305659538, −24.42899106750032366622365338122, −23.50110978599159903262219635098, −22.13282820939056471581938677809, −21.71252731853658343291136826030, −20.72700007134721001835799882905, −19.88920918101885487443357727432, −19.08360904964428907077185957082, −17.76798020889784544889523850423, −16.43025381376974623048070267500, −15.57779428796228212733650887350, −14.65946993766672624221702312235, −13.71736654279585401890530401052, −13.0572175594521221711838840912, −12.40214113461530416350254868081, −10.51590390070136329189925809876, −9.94055635253347562682703007213, −8.66790647783558799770730599727, −7.262977147055241876330133233251, −6.41234834075900739529236845046, −5.18484569079420316679890278742, −3.97563729598783409005895800033, −2.62592022991136846702814436964, −2.173832101516893019603398317233,
2.173832101516893019603398317233, 2.62592022991136846702814436964, 3.97563729598783409005895800033, 5.18484569079420316679890278742, 6.41234834075900739529236845046, 7.262977147055241876330133233251, 8.66790647783558799770730599727, 9.94055635253347562682703007213, 10.51590390070136329189925809876, 12.40214113461530416350254868081, 13.0572175594521221711838840912, 13.71736654279585401890530401052, 14.65946993766672624221702312235, 15.57779428796228212733650887350, 16.43025381376974623048070267500, 17.76798020889784544889523850423, 19.08360904964428907077185957082, 19.88920918101885487443357727432, 20.72700007134721001835799882905, 21.71252731853658343291136826030, 22.13282820939056471581938677809, 23.50110978599159903262219635098, 24.42899106750032366622365338122, 25.23900430848409851967305659538, 25.93121454419649980103844702736
