| L(s) = 1 | + 1.04·2-s − 0.531·3-s − 0.902·4-s − 0.556·6-s − 2.74·7-s − 3.04·8-s − 2.71·9-s + 0.832·11-s + 0.479·12-s − 0.610·13-s − 2.87·14-s − 1.37·16-s + 4.83·17-s − 2.84·18-s + 1.45·21-s + 0.872·22-s + 3.75·23-s + 1.61·24-s − 0.639·26-s + 3.03·27-s + 2.47·28-s + 3.97·29-s + 6.92·31-s + 4.63·32-s − 0.442·33-s + 5.06·34-s + 2.45·36-s + ⋯ |
| L(s) = 1 | + 0.740·2-s − 0.306·3-s − 0.451·4-s − 0.227·6-s − 1.03·7-s − 1.07·8-s − 0.905·9-s + 0.251·11-s + 0.138·12-s − 0.169·13-s − 0.767·14-s − 0.344·16-s + 1.17·17-s − 0.670·18-s + 0.317·21-s + 0.185·22-s + 0.782·23-s + 0.329·24-s − 0.125·26-s + 0.584·27-s + 0.467·28-s + 0.738·29-s + 1.24·31-s + 0.819·32-s − 0.0770·33-s + 0.869·34-s + 0.408·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 3 | \( 1 + 0.531T + 3T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 0.832T + 11T^{2} \) |
| 13 | \( 1 + 0.610T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.677T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 + 4.97T + 83T^{2} \) |
| 89 | \( 1 - 5.88T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.15098560691714232037666710546, −6.45987891068929104347746779326, −5.97897886710532068649941078557, −5.23689362646747094538794643248, −4.79134493408937998543531483729, −3.72269779066373461515564687877, −3.21655876380915029817282853633, −2.62946198119653602707935911705, −1.01904467469397438568838597693, 0,
1.01904467469397438568838597693, 2.62946198119653602707935911705, 3.21655876380915029817282853633, 3.72269779066373461515564687877, 4.79134493408937998543531483729, 5.23689362646747094538794643248, 5.97897886710532068649941078557, 6.45987891068929104347746779326, 7.15098560691714232037666710546
