L(s) = 1 | + 1.73i·5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s − 1.99·25-s + (−0.5 + 0.866i)29-s + (−1.5 − 0.866i)37-s + (1.5 + 0.866i)41-s + (1.49 − 0.866i)45-s + (0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)61-s + (1.49 + 0.866i)65-s + 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯ |
L(s) = 1 | + 1.73i·5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s − 1.99·25-s + (−0.5 + 0.866i)29-s + (−1.5 − 0.866i)37-s + (1.5 + 0.866i)41-s + (1.49 − 0.866i)45-s + (0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)61-s + (1.49 + 0.866i)65-s + 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6888869348\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6888869348\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−12.66946801926131138253777258696, −11.46495287541782733866618194983, −10.89785009441896763556624464775, −9.929997555811770108372414413849, −8.802294370848369744271889143508, −7.44019572724384792121082795998, −6.63249688970102400403475350525, −5.62967791504167381176092017247, −3.65631048088994838868633934788, −2.73420764988430395564816193588,
1.83550410882520121517010296642, 4.09371039513535875014140741300, 5.05956358824527363159912660992, 6.13426096303159739593014346481, 7.80720753815290391350990065629, 8.643663944488015751305887033254, 9.310941861310981309639584571100, 10.68980671934087732052072348512, 11.68685972121358948869191361955, 12.60393920561146789535734899655