| L(s) = 1 | + 2-s − 1.73·3-s − 1.73·5-s − 1.73·6-s − 7-s − 8-s + 1.99·9-s − 1.73·10-s − 11-s − 14-s + 2.99·15-s − 16-s − 1.73·17-s + 1.99·18-s + 1.73·21-s − 22-s − 2·23-s + 1.73·24-s + 1.99·25-s − 1.73·27-s − 29-s + 2.99·30-s + 1.73·33-s − 1.73·34-s + 1.73·35-s + 1.73·40-s + 1.73·41-s + ⋯ |
| L(s) = 1 | + 2-s − 1.73·3-s − 1.73·5-s − 1.73·6-s − 7-s − 8-s + 1.99·9-s − 1.73·10-s − 11-s − 14-s + 2.99·15-s − 16-s − 1.73·17-s + 1.99·18-s + 1.73·21-s − 22-s − 2·23-s + 1.73·24-s + 1.99·25-s − 1.73·27-s − 29-s + 2.99·30-s + 1.73·33-s − 1.73·34-s + 1.73·35-s + 1.73·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1038868474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1038868474\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 2T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 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
−8.782478952760063691003021097310, −7.75178268885156786753077014015, −7.11746300116069193607360004874, −6.21834174073620449732068242903, −5.84251601188439348149896939969, −4.85533111177461812743520028068, −4.22075374901353123559614346399, −3.84137852850043809542149813079, −2.60974997414010456890227051542, −0.24613183546964114335783284909,
0.24613183546964114335783284909, 2.60974997414010456890227051542, 3.84137852850043809542149813079, 4.22075374901353123559614346399, 4.85533111177461812743520028068, 5.84251601188439348149896939969, 6.21834174073620449732068242903, 7.11746300116069193607360004874, 7.75178268885156786753077014015, 8.782478952760063691003021097310
