| L(s) = 1 | + 2·2-s + 10·3-s + 4·4-s − 6·5-s + 20·6-s − 10·7-s + 8·8-s + 73·9-s − 12·10-s − 54·11-s + 40·12-s − 82·13-s − 20·14-s − 60·15-s + 16·16-s + 42·17-s + 146·18-s + 134·19-s − 24·20-s − 100·21-s − 108·22-s + 48·23-s + 80·24-s − 89·25-s − 164·26-s + 460·27-s − 40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 1/2·4-s − 0.536·5-s + 1.36·6-s − 0.539·7-s + 0.353·8-s + 2.70·9-s − 0.379·10-s − 1.48·11-s + 0.962·12-s − 1.74·13-s − 0.381·14-s − 1.03·15-s + 1/4·16-s + 0.599·17-s + 1.91·18-s + 1.61·19-s − 0.268·20-s − 1.03·21-s − 1.04·22-s + 0.435·23-s + 0.680·24-s − 0.711·25-s − 1.23·26-s + 3.27·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.290404485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.290404485\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 41 | \( 1 - p T \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 134 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 390 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 38 T + p^{3} T^{2} \) |
| 67 | \( 1 + 610 T + p^{3} T^{2} \) |
| 71 | \( 1 + 42 T + p^{3} T^{2} \) |
| 73 | \( 1 - 110 T + p^{3} T^{2} \) |
| 79 | \( 1 - 950 T + p^{3} T^{2} \) |
| 83 | \( 1 - 900 T + p^{3} T^{2} \) |
| 89 | \( 1 - 138 T + p^{3} T^{2} \) |
| 97 | \( 1 - 170 T + p^{3} T^{2} \) |
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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
−13.88925582678155027172121797733, −13.01715734050641946733876906298, −12.16957823763697448076827295603, −10.23165757573874958538882566309, −9.390868976068457134587438022761, −7.70246473570802112591651630335, −7.45517422870250577199208867013, −4.98312086715730887913075033181, −3.42453634987381401930674881353, −2.52865758156178631512401642655,
2.52865758156178631512401642655, 3.42453634987381401930674881353, 4.98312086715730887913075033181, 7.45517422870250577199208867013, 7.70246473570802112591651630335, 9.390868976068457134587438022761, 10.23165757573874958538882566309, 12.16957823763697448076827295603, 13.01715734050641946733876906298, 13.88925582678155027172121797733
