L(s) = 1 | + i·2-s − i·3-s − 4-s − 0.837i·5-s + 6-s − i·8-s − 9-s + 0.837·10-s + (0.697 − 3.24i)11-s + i·12-s + 2.59·13-s − 0.837·15-s + 16-s + 5.97·17-s − i·18-s − 3.11·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.374i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.264·10-s + (0.210 − 0.977i)11-s + 0.288i·12-s + 0.719·13-s − 0.216·15-s + 0.250·16-s + 1.44·17-s − 0.235i·18-s − 0.713·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.647336936\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647336936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.697 + 3.24i)T \) |
good | 5 | \( 1 + 0.837iT - 5T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 5.38iT - 29T^{2} \) |
| 31 | \( 1 + 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 1.27iT - 43T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 9.68iT - 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + 8.40iT - 89T^{2} \) |
| 97 | \( 1 + 0.786iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.406892063220956125952575493527, −7.909050230118605014722963850852, −6.93094642245553641262496715107, −6.34135104571286379615527398192, −5.61181039317166959073881037406, −4.95493211923927642448926652164, −3.77866400901678804919669614487, −3.07244794951625379134180191500, −1.55988140387697961364971953728, −0.58033367246852718681148359998,
1.20562731012024759617247272530, 2.27849741896284531121492285180, 3.42547817296306050441622592102, 3.76774398568307565958347655258, 5.00619521796640476110546494231, 5.36214407811775396897231675461, 6.67162248085925426698089729630, 7.16923946605742512343508044686, 8.443108568482292423076016396217, 8.738737037072673697653912318401