L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 25-s − 26-s − 28-s − 29-s + 31-s − 32-s − 34-s − 35-s − 37-s + 38-s − 40-s − 41-s − 43-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 25-s − 26-s − 28-s − 29-s + 31-s − 32-s − 34-s − 35-s − 37-s + 38-s − 40-s − 41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6988237658\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6988237658\) |
\(L(1)\) |
\(\approx\) |
\(0.7746280617\) |
\(L(1)\) |
\(\approx\) |
\(0.7746280617\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 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
−32.21762384943363625140375589753, −30.18478233071195498047850088711, −29.61957153017785415189247135551, −28.473712795822786583811835280959, −27.674486428848214865235478006793, −26.11883667105154450563610395343, −25.572435838559562173395108703657, −24.6593374485508516502684605759, −23.02537998104311620958290847568, −21.642424019123861664425655738380, −20.61805164647212371890821188456, −19.33905554671463660455604473623, −18.45088290621407564474621445322, −17.13009762136880476636778070362, −16.45807873837363079437020244799, −14.99023247855555059660755042503, −13.483043179299534065516900783939, −12.109009320822192739030152644179, −10.555819577806464845866994174001, −9.600206982670612259211105489811, −8.6100294925213863239155525085, −6.75267169956386058832091004744, −5.95435910875393137127761845837, −3.316091983481942862003250631661, −1.57186578602810803483908737273,
1.57186578602810803483908737273, 3.316091983481942862003250631661, 5.95435910875393137127761845837, 6.75267169956386058832091004744, 8.6100294925213863239155525085, 9.600206982670612259211105489811, 10.555819577806464845866994174001, 12.109009320822192739030152644179, 13.483043179299534065516900783939, 14.99023247855555059660755042503, 16.45807873837363079437020244799, 17.13009762136880476636778070362, 18.45088290621407564474621445322, 19.33905554671463660455604473623, 20.61805164647212371890821188456, 21.642424019123861664425655738380, 23.02537998104311620958290847568, 24.6593374485508516502684605759, 25.572435838559562173395108703657, 26.11883667105154450563610395343, 27.674486428848214865235478006793, 28.473712795822786583811835280959, 29.61957153017785415189247135551, 30.18478233071195498047850088711, 32.21762384943363625140375589753