| L(s) = 1 | + 1.73·3-s + 5-s + 1.99·9-s − 1.73·11-s − 13-s + 1.73·15-s + 1.73·27-s + 29-s + 1.73·31-s − 2.99·33-s − 1.73·39-s − 1.73·43-s + 1.99·45-s − 1.73·47-s + 49-s − 53-s − 1.73·55-s − 65-s − 1.73·79-s + 0.999·81-s + 1.73·87-s + 2.99·93-s − 3.46·99-s + 109-s − 1.99·117-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 5-s + 1.99·9-s − 1.73·11-s − 13-s + 1.73·15-s + 1.73·27-s + 29-s + 1.73·31-s − 2.99·33-s − 1.73·39-s − 1.73·43-s + 1.99·45-s − 1.73·47-s + 49-s − 53-s − 1.73·55-s − 65-s − 1.73·79-s + 0.999·81-s + 1.73·87-s + 2.99·93-s − 3.46·99-s + 109-s − 1.99·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.152118726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152118726\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.73T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 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
−9.533601622857176110276155635692, −8.479756662247997008727690280096, −8.091132641447739148991550885598, −7.31618417075113247649406611278, −6.39038375249028423483166925177, −5.20242877427505774346651151348, −4.53363505654224886975326666487, −3.08397016207228169592284343951, −2.63620387155371426573231115602, −1.80143124506088055118383240600,
1.80143124506088055118383240600, 2.63620387155371426573231115602, 3.08397016207228169592284343951, 4.53363505654224886975326666487, 5.20242877427505774346651151348, 6.39038375249028423483166925177, 7.31618417075113247649406611278, 8.091132641447739148991550885598, 8.479756662247997008727690280096, 9.533601622857176110276155635692
