| L(s) = 1 | + (−0.360 − 0.706i)2-s + (1.40 + 1.00i)3-s + (0.805 − 1.10i)4-s + (−0.241 − 1.52i)5-s + (0.203 − 1.35i)6-s + (−0.678 + 0.579i)7-s + (−2.64 − 0.418i)8-s + (0.973 + 2.83i)9-s + (−0.992 + 0.720i)10-s + (1.38 + 0.848i)11-s + (2.25 − 0.752i)12-s + (1.62 − 0.127i)13-s + (0.654 + 0.271i)14-s + (1.19 − 2.39i)15-s + (−0.191 − 0.590i)16-s + (−2.52 + 0.605i)17-s + ⋯ |
| L(s) = 1 | + (−0.254 − 0.499i)2-s + (0.813 + 0.581i)3-s + (0.402 − 0.554i)4-s + (−0.108 − 0.682i)5-s + (0.0832 − 0.554i)6-s + (−0.256 + 0.219i)7-s + (−0.933 − 0.147i)8-s + (0.324 + 0.945i)9-s + (−0.313 + 0.227i)10-s + (0.417 + 0.255i)11-s + (0.650 − 0.217i)12-s + (0.450 − 0.0354i)13-s + (0.174 + 0.0724i)14-s + (0.308 − 0.618i)15-s + (−0.0479 − 0.147i)16-s + (−0.611 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14878 - 0.407861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14878 - 0.407861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.40 - 1.00i)T \) |
| 41 | \( 1 + (4.98 + 4.01i)T \) |
| good | 2 | \( 1 + (0.360 + 0.706i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.241 + 1.52i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (0.678 - 0.579i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 0.848i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-1.62 + 0.127i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (2.52 - 0.605i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (3.92 + 0.308i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (2.26 - 6.95i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.22 - 0.533i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-0.573 - 0.788i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (5.57 + 4.05i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (7.93 - 4.04i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-5.50 + 6.44i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.59 + 6.64i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-8.26 - 2.68i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.29 + 10.3i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-1.55 + 0.954i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-7.83 + 12.7i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-7.88 - 7.88i)T + 73iT^{2} \) |
| 79 | \( 1 + (-0.996 + 0.412i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 3.29iT - 83T^{2} \) |
| 89 | \( 1 + (-5.01 - 5.87i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (8.07 + 13.1i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−13.30961939519854294622973870315, −12.21230078677902045187365198768, −11.06899768578711547021744771910, −10.09383576639618021094390133155, −9.175715118088633140766838684492, −8.419633318407780564076541583823, −6.71386252586533492403516627184, −5.18197599195014493415921255306, −3.65491110481498611642070075971, −1.96295739468817831972893995817,
2.55820865951848214824470107830, 3.80390573604419229047784533344, 6.52383426324014942276376281691, 6.81922165895074742750078766417, 8.212940240655392433054006073112, 8.824195913120226289715069471433, 10.37111986214848010880152564665, 11.62485559249655699149295182783, 12.61319430107612117140906159014, 13.61356478679875987446465452664
