| L(s) = 1 | − 4.87·3-s − 5·5-s + 7·7-s − 3.25·9-s + 49.7·11-s + 42.7·13-s + 24.3·15-s + 27.2·17-s − 97.8·19-s − 34.1·21-s − 59.1·23-s + 25·25-s + 147.·27-s − 99.2·29-s + 127.·31-s − 242.·33-s − 35·35-s − 47.9·37-s − 208.·39-s + 169.·41-s − 436.·43-s + 16.2·45-s − 427.·47-s + 49·49-s − 132.·51-s + 406.·53-s − 248.·55-s + ⋯ |
| L(s) = 1 | − 0.937·3-s − 0.447·5-s + 0.377·7-s − 0.120·9-s + 1.36·11-s + 0.912·13-s + 0.419·15-s + 0.388·17-s − 1.18·19-s − 0.354·21-s − 0.536·23-s + 0.200·25-s + 1.05·27-s − 0.635·29-s + 0.738·31-s − 1.27·33-s − 0.169·35-s − 0.212·37-s − 0.856·39-s + 0.647·41-s − 1.54·43-s + 0.0538·45-s − 1.32·47-s + 0.142·49-s − 0.364·51-s + 1.05·53-s − 0.609·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.300206068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300206068\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 4.87T + 27T^{2} \) |
| 11 | \( 1 - 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 47.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 427.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 730.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 463.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 421.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 845.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 423.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−9.420405736269646679914071455576, −8.568609525754151458296732806520, −7.910661569962277002213301094303, −6.54953529754033746686548141909, −6.30455694733428474116222868592, −5.19372260117373778774081905350, −4.26006696926155811414211022055, −3.42209136336495592617516466835, −1.76855685313115880701346186365, −0.64004712433972545739587737205,
0.64004712433972545739587737205, 1.76855685313115880701346186365, 3.42209136336495592617516466835, 4.26006696926155811414211022055, 5.19372260117373778774081905350, 6.30455694733428474116222868592, 6.54953529754033746686548141909, 7.910661569962277002213301094303, 8.568609525754151458296732806520, 9.420405736269646679914071455576
