L(s) = 1 | + (1.71 + 0.222i)3-s + (0.234 + 0.405i)5-s + (−0.212 + 2.63i)7-s + (2.90 + 0.764i)9-s + (−0.674 + 1.16i)11-s + (−3.16 + 5.48i)13-s + (0.311 + 0.748i)15-s + (−2.47 − 4.28i)17-s + (−2.38 + 4.13i)19-s + (−0.951 + 4.48i)21-s + (3.81 + 6.60i)23-s + (2.39 − 4.14i)25-s + (4.81 + 1.95i)27-s + (−1.80 − 3.12i)29-s − 6.49·31-s + ⋯ |
L(s) = 1 | + (0.991 + 0.128i)3-s + (0.104 + 0.181i)5-s + (−0.0802 + 0.996i)7-s + (0.966 + 0.254i)9-s + (−0.203 + 0.352i)11-s + (−0.877 + 1.52i)13-s + (0.0805 + 0.193i)15-s + (−0.599 − 1.03i)17-s + (−0.548 + 0.949i)19-s + (−0.207 + 0.978i)21-s + (0.795 + 1.37i)23-s + (0.478 − 0.828i)25-s + (0.926 + 0.377i)27-s + (−0.335 − 0.580i)29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.041754648\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041754648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 - 0.222i)T \) |
| 7 | \( 1 + (0.212 - 2.63i)T \) |
good | 5 | \( 1 + (-0.234 - 0.405i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.674 - 1.16i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.16 - 5.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.47 + 4.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.81 - 6.60i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.80 + 3.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0251 - 0.0435i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.431 - 0.748i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + (-5.84 - 10.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 + (4.90 + 8.49i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.30 + 9.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.97 - 12.0i)T + (-48.5 + 84.0i)T^{2} \) |
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
−9.810165739660487632939187757682, −9.231928557223493296718694533955, −8.814468547987417628856370082279, −7.53314409986252963432778149482, −7.08883551818050551579550681167, −5.87187009055070573617428521059, −4.77741608894895149630917200943, −3.90042302877353684589003370776, −2.54853297101052596346598929395, −2.02775302740984073404713197937,
0.819565711309638034780496160771, 2.39492312491742103200457733722, 3.32568865970065779918522004268, 4.34332121098379763133721196315, 5.28838857426061131384815790941, 6.70510871041028088432089297940, 7.27390533868305093593475767481, 8.230765023911957528268556654092, 8.767828490399514546659337360264, 9.778115061987787502448561801423