| L(s) = 1 | + i·3-s + (−2.19 − 0.440i)5-s + 1.54i·7-s − 9-s − 4.84·11-s − 0.136i·13-s + (0.440 − 2.19i)15-s − i·17-s − 8.51·19-s − 1.54·21-s − 0.206i·23-s + (4.61 + 1.93i)25-s − i·27-s − 4.31·29-s + 4.24·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.980 − 0.197i)5-s + 0.583i·7-s − 0.333·9-s − 1.45·11-s − 0.0378i·13-s + (0.113 − 0.566i)15-s − 0.242i·17-s − 1.95·19-s − 0.336·21-s − 0.0430i·23-s + (0.922 + 0.386i)25-s − 0.192i·27-s − 0.800·29-s + 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7264598980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7264598980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.19 + 0.440i)T \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.54iT - 7T^{2} \) |
| 11 | \( 1 + 4.84T + 11T^{2} \) |
| 13 | \( 1 + 0.136iT - 13T^{2} \) |
| 19 | \( 1 + 8.51T + 19T^{2} \) |
| 23 | \( 1 + 0.206iT - 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 1.78iT - 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 4.77iT - 43T^{2} \) |
| 47 | \( 1 - 0.302iT - 47T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 3.85iT - 67T^{2} \) |
| 71 | \( 1 - 5.16T + 71T^{2} \) |
| 73 | \( 1 + 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 - 3.45iT - 97T^{2} \) |
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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
−8.330877938421510965543054162900, −7.951101543113258493513430728015, −7.01428729619999681862740715219, −6.10498083911547919481595933143, −5.26612205927288654663865716153, −4.63675360106103834938550228769, −3.91432248293250325731882576050, −2.93699720377715836367240093541, −2.18394676389384453522491048647, −0.34510645754054967910077409050,
0.61476696709494572744937641785, 2.10223496095909557039304851720, 2.88544857802712963697573441710, 3.95438797417588169691413285933, 4.52178895980476528514488856813, 5.56270244959209204036980020261, 6.36923724959197477505201987182, 7.20992872443524079443509465035, 7.63367299469220122219465789782, 8.332031768118362241361307709501
