L(s) = 1 | + 1.73·5-s + 13-s − 1.73·17-s + 1.99·25-s − 1.73·29-s − 37-s + 49-s − 61-s + 1.73·65-s + 73-s − 2.99·85-s + 1.73·89-s − 2·97-s + 109-s − 1.73·113-s + ⋯ |
L(s) = 1 | + 1.73·5-s + 13-s − 1.73·17-s + 1.99·25-s − 1.73·29-s − 37-s + 49-s − 61-s + 1.73·65-s + 73-s − 2.99·85-s + 1.73·89-s − 2·97-s + 109-s − 1.73·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.405537856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405537856\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 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 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.721169801455554657914560873212, −9.099888208310593188301189230783, −8.549164322737047276677506946624, −7.20839705442414954117809075435, −6.39122929017153888476325403749, −5.80628068167182434272873046839, −4.93835737667782121868564297420, −3.75572639556755076539088248769, −2.41836033827084064144893377090, −1.62511940481098247442134060471,
1.62511940481098247442134060471, 2.41836033827084064144893377090, 3.75572639556755076539088248769, 4.93835737667782121868564297420, 5.80628068167182434272873046839, 6.39122929017153888476325403749, 7.20839705442414954117809075435, 8.549164322737047276677506946624, 9.099888208310593188301189230783, 9.721169801455554657914560873212