| L(s) = 1 | + 1.73·2-s − 3-s + 1.99·4-s + 5-s − 1.73·6-s + 1.73·8-s + 9-s + 1.73·10-s − 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s + 1.99·20-s + 23-s − 1.73·24-s + 25-s − 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s + 1.73·40-s + 45-s + 1.73·46-s − 47-s − 0.999·48-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 3-s + 1.99·4-s + 5-s − 1.73·6-s + 1.73·8-s + 9-s + 1.73·10-s − 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s + 1.99·20-s + 23-s − 1.73·24-s + 25-s − 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s + 1.73·40-s + 45-s + 1.73·46-s − 47-s − 0.999·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.602238298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.602238298\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.73T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.656159348438347892927595392056, −8.778033178593796715735826706120, −7.20327146837067514980066063956, −6.69962481250731999581600669534, −6.06243175712680702213385968073, −5.30067903487659601179785626884, −4.76751469542819473074050634909, −3.90850182961347727927371126908, −2.66212566692108012495035884463, −1.70759946372457092048555556385,
1.70759946372457092048555556385, 2.66212566692108012495035884463, 3.90850182961347727927371126908, 4.76751469542819473074050634909, 5.30067903487659601179785626884, 6.06243175712680702213385968073, 6.69962481250731999581600669534, 7.20327146837067514980066063956, 8.778033178593796715735826706120, 9.656159348438347892927595392056
