| L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (4.33 − 0.414i)5-s + (−0.989 + 0.142i)6-s + (−0.705 + 2.55i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−1.02 + 4.23i)10-s + (−1.85 + 0.642i)11-s + (0.189 − 0.981i)12-s + (0.782 + 2.66i)13-s + (−2.17 − 1.50i)14-s + (2.35 + 3.66i)15-s + (0.235 + 0.971i)16-s + (−2.37 − 0.948i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (1.94 − 0.185i)5-s + (−0.404 + 0.0580i)6-s + (−0.266 + 0.963i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.325 + 1.33i)10-s + (−0.559 + 0.193i)11-s + (0.0546 − 0.283i)12-s + (0.217 + 0.739i)13-s + (−0.582 − 0.401i)14-s + (0.608 + 0.946i)15-s + (0.0589 + 0.242i)16-s + (−0.574 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874823 + 1.61729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874823 + 1.61729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.327 - 0.945i)T \) |
| 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (0.705 - 2.55i)T \) |
| 23 | \( 1 + (2.66 - 3.99i)T \) |
| good | 5 | \( 1 + (-4.33 + 0.414i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (1.85 - 0.642i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-0.782 - 2.66i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (2.37 + 0.948i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (4.71 - 1.88i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.347 + 2.41i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-9.27 + 0.441i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (1.15 + 0.820i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-6.95 - 3.17i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.85 - 10.6i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-11.4 + 6.63i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.49 - 2.61i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-9.61 - 2.33i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (5.61 + 2.89i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (2.25 + 11.6i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (6.03 + 6.96i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-4.58 + 5.83i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (0.301 - 0.316i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.84 - 6.23i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.375 + 7.88i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (2.91 - 6.38i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−9.985138097001628441855095377943, −9.380263161376252330897982991127, −8.882207993690590585045718940887, −8.023168980190962896349548010657, −6.52176383027490568350113491539, −6.10324049596044939739714589353, −5.29330486989065402881397754254, −4.42573093433017235318756930150, −2.66271270201117743623778166800, −1.86358942217964868213411375255,
0.894247899302641194338930203882, 2.19292039760019595390499300581, 2.82436493046125317008119755098, 4.25877090595263230894564459491, 5.49882432559526161815319289889, 6.37133337429789082528113501913, 7.08409291545192247716584181672, 8.359630417504363317224180205035, 8.936047419596876581473735185994, 10.05353810807772084979802735905
