| L(s) = 1 | + 4·5-s + 2·7-s − 4·11-s + 13-s − 2·17-s + 2·19-s + 11·25-s + 6·29-s + 10·31-s + 8·35-s + 10·37-s − 8·41-s − 4·43-s − 4·47-s − 3·49-s + 10·53-s − 16·55-s − 8·59-s − 14·61-s + 4·65-s − 2·67-s + 16·71-s − 10·73-s − 8·77-s + 16·79-s − 8·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 11/5·25-s + 1.11·29-s + 1.79·31-s + 1.35·35-s + 1.64·37-s − 1.24·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s − 2.15·55-s − 1.04·59-s − 1.79·61-s + 0.496·65-s − 0.244·67-s + 1.89·71-s − 1.17·73-s − 0.911·77-s + 1.80·79-s − 0.867·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620154302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620154302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.323452616240301180517204990354, −8.437984313060842025739365652378, −7.82743937659030627971434697795, −6.60471812809635507315319476307, −6.07506199316679251481919750358, −5.10290917264965015136465501487, −4.70084323749332305530012565841, −2.96541597463640709898029251011, −2.25446002723202452780259837823, −1.20578006642341497703303933331,
1.20578006642341497703303933331, 2.25446002723202452780259837823, 2.96541597463640709898029251011, 4.70084323749332305530012565841, 5.10290917264965015136465501487, 6.07506199316679251481919750358, 6.60471812809635507315319476307, 7.82743937659030627971434697795, 8.437984313060842025739365652378, 9.323452616240301180517204990354
