| L(s) = 1 | − 2-s + 1.53·3-s + 4-s − 1.53·6-s − 2.87·7-s − 8-s − 0.630·9-s − 1.09·11-s + 1.53·12-s + 4.53·13-s + 2.87·14-s + 16-s + 2.80·17-s + 0.630·18-s − 5.04·19-s − 4.43·21-s + 1.09·22-s + 7.41·23-s − 1.53·24-s − 4.53·26-s − 5.58·27-s − 2.87·28-s + 6.68·29-s + 3.51·31-s − 32-s − 1.68·33-s − 2.80·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.888·3-s + 0.5·4-s − 0.628·6-s − 1.08·7-s − 0.353·8-s − 0.210·9-s − 0.329·11-s + 0.444·12-s + 1.25·13-s + 0.769·14-s + 0.250·16-s + 0.679·17-s + 0.148·18-s − 1.15·19-s − 0.967·21-s + 0.232·22-s + 1.54·23-s − 0.314·24-s − 0.890·26-s − 1.07·27-s − 0.544·28-s + 1.24·29-s + 0.630·31-s − 0.176·32-s − 0.292·33-s − 0.480·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464416312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464416312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 7.41T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 6.29T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 + 6.51T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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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
−9.174884464980807358483994496019, −8.438100824807432969936853997324, −8.069068671636531002664992776468, −6.86172017737696628188444217001, −6.36023965416904923034384284052, −5.36333268517365334261258117641, −3.89638786019622706403047285405, −3.12326921126962668452747384709, −2.40300268901560899349093291589, −0.872344792346611976073525024946,
0.872344792346611976073525024946, 2.40300268901560899349093291589, 3.12326921126962668452747384709, 3.89638786019622706403047285405, 5.36333268517365334261258117641, 6.36023965416904923034384284052, 6.86172017737696628188444217001, 8.069068671636531002664992776468, 8.438100824807432969936853997324, 9.174884464980807358483994496019
