| L(s) = 1 | − 2-s − 3-s + 4-s − 1.73·5-s + 6-s − 7-s − 8-s + 1.73·10-s − 1.73·11-s − 12-s + 14-s + 1.73·15-s + 16-s − 1.73·20-s + 21-s + 1.73·22-s + 24-s + 1.99·25-s + 27-s − 28-s − 1.73·30-s − 32-s + 1.73·33-s + 1.73·35-s − 37-s + 1.73·40-s + 1.73·41-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 1.73·5-s + 6-s − 7-s − 8-s + 1.73·10-s − 1.73·11-s − 12-s + 14-s + 1.73·15-s + 16-s − 1.73·20-s + 21-s + 1.73·22-s + 24-s + 1.99·25-s + 27-s − 28-s − 1.73·30-s − 32-s + 1.73·33-s + 1.73·35-s − 37-s + 1.73·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1316 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1316 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1356177646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1356177646\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 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
−10.13481816248484804024072920009, −8.863902107607138567316593555053, −8.177953014235833974434445376789, −7.45052908304830092134719898513, −6.79536403747614977910585987053, −5.82738769291136303910028854965, −4.91898772334083797436014514088, −3.54378054825937984262985688391, −2.71599597010892613243063274462, −0.44805428017341995693368082125,
0.44805428017341995693368082125, 2.71599597010892613243063274462, 3.54378054825937984262985688391, 4.91898772334083797436014514088, 5.82738769291136303910028854965, 6.79536403747614977910585987053, 7.45052908304830092134719898513, 8.177953014235833974434445376789, 8.863902107607138567316593555053, 10.13481816248484804024072920009
