L(s) = 1 | − 2-s + 4-s + (1.5 + 2.59i)5-s + (2.5 − 0.866i)7-s − 8-s + (−1.5 − 2.59i)10-s + (−1.5 + 2.59i)11-s + (−1 + 1.73i)13-s + (−2.5 + 0.866i)14-s + 16-s + (3 + 5.19i)17-s + (−1 + 1.73i)19-s + (1.5 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−3 − 5.19i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.670 + 1.16i)5-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.474 − 0.821i)10-s + (−0.452 + 0.783i)11-s + (−0.277 + 0.480i)13-s + (−0.668 + 0.231i)14-s + 0.250·16-s + (0.727 + 1.26i)17-s + (−0.229 + 0.397i)19-s + (0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.300254646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.300254646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.26980607198451944229242033133, −9.332961177908088222805515235009, −8.268844182903131002529243359122, −7.62184572398253721840102770906, −6.80973395041526012848909616557, −6.07009735323186838577893313892, −4.95863252746363072736864922124, −3.73695121283881014905113206878, −2.39150605009662662378889033192, −1.66039774469326545763691846884,
0.72350189192768568880147182121, 1.85763538174346659118100126300, 3.02531344070315714082326889820, 4.69349823285998869070588357282, 5.37767761339440120497449487034, 6.05326367348329007999016342382, 7.50517691809524325768051837531, 8.088168891708433689094398377918, 8.775262187367468271700633997296, 9.569685192289642502328349543342