| L(s) = 1 | + 2-s + 4-s − 5-s + 1.73·7-s + 8-s − 10-s − 1.73·13-s + 1.73·14-s + 16-s + 19-s − 20-s + 23-s + 25-s − 1.73·26-s + 1.73·28-s + 32-s − 1.73·35-s + 38-s − 40-s − 1.73·41-s + 46-s − 47-s + 1.99·49-s + 50-s − 1.73·52-s − 53-s + 1.73·56-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 1.73·7-s + 8-s − 10-s − 1.73·13-s + 1.73·14-s + 16-s + 19-s − 20-s + 23-s + 25-s − 1.73·26-s + 1.73·28-s + 32-s − 1.73·35-s + 38-s − 40-s − 1.73·41-s + 46-s − 47-s + 1.99·49-s + 50-s − 1.73·52-s − 53-s + 1.73·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.434081938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434081938\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - 1.73T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 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
−8.487711935563782900956867445667, −7.88339696295926010696013236388, −7.31857335548969746701501841195, −6.78212316264563716005023414725, −5.22944657780010103416467180690, −5.07087499591302126826955994498, −4.38526473926340885184688233917, −3.38558437304163758272421347164, −2.47743341531067624936286404992, −1.37798830320113286732611904974,
1.37798830320113286732611904974, 2.47743341531067624936286404992, 3.38558437304163758272421347164, 4.38526473926340885184688233917, 5.07087499591302126826955994498, 5.22944657780010103416467180690, 6.78212316264563716005023414725, 7.31857335548969746701501841195, 7.88339696295926010696013236388, 8.487711935563782900956867445667
