L(s) = 1 | + 1.73·5-s + 7-s − 1.73·11-s − 1.73·17-s − 19-s + 1.99·25-s + 1.73·35-s − 43-s + 1.73·47-s − 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯ |
L(s) = 1 | + 1.73·5-s + 7-s − 1.73·11-s − 1.73·17-s − 19-s + 1.99·25-s + 1.73·35-s − 43-s + 1.73·47-s − 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.187721470\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.187721470\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 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
−10.63819076265861274386595416207, −9.986579019578883302601213491787, −8.906192865166512475222754760476, −8.292569501938580457560570263668, −7.09915660721554118584910851156, −6.10394053929476773795301998781, −5.27195744699599578733541998541, −4.54795630381886956161819096058, −2.54548297162908671539998415590, −1.94363267413058732322543134869,
1.94363267413058732322543134869, 2.54548297162908671539998415590, 4.54795630381886956161819096058, 5.27195744699599578733541998541, 6.10394053929476773795301998781, 7.09915660721554118584910851156, 8.292569501938580457560570263668, 8.906192865166512475222754760476, 9.986579019578883302601213491787, 10.63819076265861274386595416207