| L(s) = 1 | + (1.40 + 0.140i)2-s + (0.755 − 0.654i)3-s + (1.96 + 0.395i)4-s + (−3.07 + 0.442i)5-s + (1.15 − 0.815i)6-s + (1.27 − 2.78i)7-s + (2.70 + 0.832i)8-s + (0.142 − 0.989i)9-s + (−4.39 + 0.189i)10-s + (0.100 − 0.340i)11-s + (1.74 − 0.984i)12-s + (5.09 − 2.32i)13-s + (2.18 − 3.74i)14-s + (−2.03 + 2.34i)15-s + (3.68 + 1.55i)16-s + (4.48 + 2.88i)17-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.0994i)2-s + (0.436 − 0.378i)3-s + (0.980 + 0.197i)4-s + (−1.37 + 0.197i)5-s + (0.471 − 0.332i)6-s + (0.481 − 1.05i)7-s + (0.955 + 0.294i)8-s + (0.0474 − 0.329i)9-s + (−1.38 + 0.0599i)10-s + (0.0301 − 0.102i)11-s + (0.502 − 0.284i)12-s + (1.41 − 0.645i)13-s + (0.583 − 1.00i)14-s + (−0.525 + 0.606i)15-s + (0.921 + 0.388i)16-s + (1.08 + 0.699i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65551 - 0.720019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65551 - 0.720019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.140i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (3.41 + 3.36i)T \) |
| good | 5 | \( 1 + (3.07 - 0.442i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.27 + 2.78i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.100 + 0.340i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.09 + 2.32i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.48 - 2.88i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.803 + 1.25i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.99 - 6.21i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.04 - 1.20i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.87 + 0.844i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.40 - 9.74i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.06 - 4.38i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + (6.17 + 2.82i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-12.5 + 5.73i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.74 + 4.98i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.617 - 2.10i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.31 + 2.44i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (6.36 - 4.09i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.12 - 11.2i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.513 - 0.0738i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (5.20 + 6.00i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.491 - 3.41i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.04200098141164646517180007544, −10.22754719868062378505427133450, −8.237906351838780165061318884393, −8.070772549673160590744611741134, −7.11916031995331354966878081604, −6.24062486406876858128656705778, −4.86020711209612733507655962085, −3.67837206640112582502102708222, −3.42498568977573661891462909003, −1.37011150793245436051864567704,
1.89603048166614306907988375599, 3.43574546073056295272609806937, 3.97799260535534301473068439437, 5.10997364465481694080967344225, 5.99974276909066418398522298614, 7.37683321433669363730022829424, 8.106552402931149082278076954572, 8.928897996356506828041535001859, 10.14610503821930108651198886051, 11.35640494110189141471762307291
