L(s) = 1 | + (−0.515 − 0.892i)5-s + (−2.63 − 0.277i)7-s + (0.792 − 1.37i)11-s + (−2.52 + 4.37i)13-s + (−2.58 − 4.47i)17-s + (−0.392 + 0.680i)19-s + (2.93 + 5.07i)23-s + (1.96 − 3.40i)25-s + (4.44 + 7.69i)29-s − 1.15·31-s + (1.10 + 2.49i)35-s + (4.07 − 7.06i)37-s + (−3.87 + 6.70i)41-s + (1.26 + 2.19i)43-s + 8.49·47-s + ⋯ |
L(s) = 1 | + (−0.230 − 0.399i)5-s + (−0.994 − 0.104i)7-s + (0.239 − 0.414i)11-s + (−0.700 + 1.21i)13-s + (−0.626 − 1.08i)17-s + (−0.0901 + 0.156i)19-s + (0.611 + 1.05i)23-s + (0.393 − 0.681i)25-s + (0.825 + 1.42i)29-s − 0.206·31-s + (0.187 + 0.421i)35-s + (0.670 − 1.16i)37-s + (−0.604 + 1.04i)41-s + (0.193 + 0.334i)43-s + 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174655553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174655553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.277i)T \) |
good | 5 | \( 1 + (0.515 + 0.892i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.792 + 1.37i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.52 - 4.37i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.392 - 0.680i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.93 - 5.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.44 - 7.69i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.87 - 6.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 2.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 + (-2.41 - 4.17i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.04 - 5.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 + (-7.12 - 12.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.69 + 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.67 + 4.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.247332851656990097571739430823, −8.523806358262134797418045892975, −7.31827346975281506681901071967, −6.90921366026835989246900323682, −6.06933494408507438787007140511, −5.01678830670439544165098089992, −4.30991802748541391686751186817, −3.31897351277874503279021946303, −2.39682957670781539918555389169, −0.906197546659440586437137710049,
0.52392702682741311351917898977, 2.29418575830759712962846517191, 3.06118579724547976904613150538, 3.99149335023996950429400039996, 4.92620900117079722682179645162, 5.96337620463161530852589654091, 6.62351154889993574396226990800, 7.30181633801727951209093701588, 8.191351226164523067661888494762, 8.928766265224725342090256563036