| L(s) = 1 | + 1.73·3-s − 5-s + 1.73·7-s + 1.99·9-s − 13-s − 1.73·15-s − 17-s + 2.99·21-s + 1.73·27-s − 1.73·35-s + 37-s − 1.73·39-s − 1.73·43-s − 1.99·45-s − 1.73·47-s + 1.99·49-s − 1.73·51-s + 3.46·63-s + 65-s + 1.73·71-s + 0.999·81-s + 85-s − 1.73·91-s − 2.99·105-s + 109-s + 1.73·111-s − 2·113-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 5-s + 1.73·7-s + 1.99·9-s − 13-s − 1.73·15-s − 17-s + 2.99·21-s + 1.73·27-s − 1.73·35-s + 37-s − 1.73·39-s − 1.73·43-s − 1.99·45-s − 1.73·47-s + 1.99·49-s − 1.73·51-s + 3.46·63-s + 65-s + 1.73·71-s + 0.999·81-s + 85-s − 1.73·91-s − 2.99·105-s + 109-s + 1.73·111-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.904233860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904233860\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 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
−9.332612742652741536054839197525, −8.478910769101201629284581610489, −8.059630376310994089045064263628, −7.59862520452270241665063849026, −6.77296438444779334904795169474, −4.99633978173547068303588507397, −4.45459838463000046231731217637, −3.62848208229560405112842363752, −2.51689124472819253401987331093, −1.71247571744736440363434726595,
1.71247571744736440363434726595, 2.51689124472819253401987331093, 3.62848208229560405112842363752, 4.45459838463000046231731217637, 4.99633978173547068303588507397, 6.77296438444779334904795169474, 7.59862520452270241665063849026, 8.059630376310994089045064263628, 8.478910769101201629284581610489, 9.332612742652741536054839197525
