L(s) = 1 | − i·2-s − 4-s + (1.22 − 2.12i)5-s + (2.62 + 0.358i)7-s + i·8-s + (−2.12 − 1.22i)10-s + (−3.67 + 2.12i)11-s + (3.62 − 2.09i)13-s + (0.358 − 2.62i)14-s + 16-s + (1.22 − 2.12i)17-s + (4.24 − 2.44i)19-s + (−1.22 + 2.12i)20-s + (2.12 + 3.67i)22-s + (5.19 + 3i)23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.547 − 0.948i)5-s + (0.990 + 0.135i)7-s + 0.353i·8-s + (−0.670 − 0.387i)10-s + (−1.10 + 0.639i)11-s + (1.00 − 0.579i)13-s + (0.0958 − 0.700i)14-s + 0.250·16-s + (0.297 − 0.514i)17-s + (0.973 − 0.561i)19-s + (−0.273 + 0.474i)20-s + (0.452 + 0.783i)22-s + (1.08 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.879725472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.879725472\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.358i)T \) |
good | 5 | \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.62 + 2.09i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.24 + 2.44i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (8.87 + 5.12i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.61iT - 31T^{2} \) |
| 37 | \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.22 + 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 + (2.15 + 1.24i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 0.717iT - 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (13.2 + 7.64i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 + (2.74 - 4.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.87 + 15.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.74 - 3.31i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.482778395533667732371141685730, −9.042255932992529307165294601033, −7.975308155674288658217987727663, −7.48115813106695332970416218090, −5.70967725506058717413121213003, −5.24561961080906486361883301925, −4.49600833475175184807257828209, −3.14513270788645663559158148829, −1.97792685605643215714491683218, −0.947834028491759467749707402237,
1.45378501618859383646333811994, 2.90007737509954269073228635719, 3.93692912924853673145816888303, 5.29056180402239894096032806175, 5.71322687577776342595042599248, 6.81341964806427915405073279257, 7.45548494210841479257079930932, 8.369730253782576359464906707086, 8.972048593959461166826164072156, 10.17401052244708908118452111924