L(s) = 1 | + (0.0242 − 0.0140i)5-s + 1.95·7-s + 2.26i·11-s + (−1.04 − 0.605i)13-s + (−5.68 + 3.27i)17-s + (−3.90 − 1.93i)19-s + (−2.86 − 1.65i)23-s + (−2.49 + 4.32i)25-s + (−0.972 + 1.68i)29-s − 7.53i·31-s + (0.0474 − 0.0274i)35-s + 1.60i·37-s + (−5.51 − 9.55i)41-s + (−6.35 − 11.0i)43-s + (5.51 + 3.18i)47-s + ⋯ |
L(s) = 1 | + (0.0108 − 0.00626i)5-s + 0.739·7-s + 0.683i·11-s + (−0.290 − 0.167i)13-s + (−1.37 + 0.795i)17-s + (−0.896 − 0.442i)19-s + (−0.597 − 0.344i)23-s + (−0.499 + 0.865i)25-s + (−0.180 + 0.312i)29-s − 1.35i·31-s + (0.00802 − 0.00463i)35-s + 0.263i·37-s + (−0.861 − 1.49i)41-s + (−0.969 − 1.67i)43-s + (0.804 + 0.464i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05889298651\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05889298651\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.90 + 1.93i)T \) |
good | 5 | \( 1 + (-0.0242 + 0.0140i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 2.26iT - 11T^{2} \) |
| 13 | \( 1 + (1.04 + 0.605i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.68 - 3.27i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.86 + 1.65i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.972 - 1.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.53iT - 31T^{2} \) |
| 37 | \( 1 - 1.60iT - 37T^{2} \) |
| 41 | \( 1 + (5.51 + 9.55i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.35 + 11.0i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.51 - 3.18i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.92 - 10.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.67 - 4.62i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.233 - 0.403i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.18 - 2.41i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.11 + 1.92i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.85 - 3.21i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.79 + 2.77i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.64iT - 83T^{2} \) |
| 89 | \( 1 + (1.78 - 3.09i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.80 + 5.65i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.040006452012101970555820940881, −8.532427120532050922353774423046, −7.68215180793265946578158181609, −7.00036621668839026213789352518, −6.17848600621935943353716643779, −5.27730265222712734518782754587, −4.44320182549100164193886069181, −3.84360693341921132615620479300, −2.34911200090482105156284555829, −1.78349024314315077649565995822,
0.01725392564743959940587374473, 1.61277889763830021158106795532, 2.51663956136452438146410340963, 3.64817664778103932704454549548, 4.60384390939288537686912247608, 5.15253774587990515749486310956, 6.34487600640361301816153949280, 6.71316378021506287616213745600, 8.001262335617580833457826724879, 8.237015569396733679349692968503