| L(s) = 1 | − 5·5-s + 0.584·7-s − 11.9·11-s + 53.7·13-s + 52.2·17-s − 144.·19-s − 54.8·23-s + 25·25-s + 201.·29-s − 217.·31-s − 2.92·35-s + 318.·37-s + 375.·41-s − 38.1·43-s − 457.·47-s − 342.·49-s + 356.·53-s + 59.6·55-s − 541.·59-s − 410.·61-s − 268.·65-s + 362.·67-s − 175.·71-s − 105.·73-s − 6.97·77-s + 357.·79-s + 1.18e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.0315·7-s − 0.326·11-s + 1.14·13-s + 0.745·17-s − 1.74·19-s − 0.497·23-s + 0.200·25-s + 1.28·29-s − 1.25·31-s − 0.0141·35-s + 1.41·37-s + 1.43·41-s − 0.135·43-s − 1.41·47-s − 0.999·49-s + 0.924·53-s + 0.146·55-s − 1.19·59-s − 0.861·61-s − 0.513·65-s + 0.660·67-s − 0.293·71-s − 0.169·73-s − 0.0103·77-s + 0.509·79-s + 1.56·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.756329121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756329121\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 0.584T + 343T^{2} \) |
| 11 | \( 1 + 11.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 54.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 318.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 375.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 38.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 541.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 410.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 362.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 175.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 357.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 64.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.896605496866918173894270262669, −8.195876936253190094943657007775, −7.63350628743787789450595651655, −6.45235314606382194235997002689, −5.96644815406462799194473642630, −4.76401955978654144099499659760, −3.99313700632277859345302293444, −3.07764555637618633000802518222, −1.87918092865390469915577135570, −0.63751683386971526895159841634,
0.63751683386971526895159841634, 1.87918092865390469915577135570, 3.07764555637618633000802518222, 3.99313700632277859345302293444, 4.76401955978654144099499659760, 5.96644815406462799194473642630, 6.45235314606382194235997002689, 7.63350628743787789450595651655, 8.195876936253190094943657007775, 8.896605496866918173894270262669
