| L(s) = 1 | + 2.84i·3-s + (0.691 + 2.12i)5-s − 0.145i·7-s − 5.10·9-s − 5.71·11-s − 5.24i·13-s + (−6.05 + 1.96i)15-s + 7.15i·17-s − 19-s + 0.412·21-s + 0.622i·23-s + (−4.04 + 2.94i)25-s − 6.00i·27-s + 5.46·29-s − 3.77·31-s + ⋯ |
| L(s) = 1 | + 1.64i·3-s + (0.309 + 0.950i)5-s − 0.0548i·7-s − 1.70·9-s − 1.72·11-s − 1.45i·13-s + (−1.56 + 0.508i)15-s + 1.73i·17-s − 0.229·19-s + 0.0901·21-s + 0.129i·23-s + (−0.808 + 0.588i)25-s − 1.15i·27-s + 1.01·29-s − 0.678·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.151603 - 0.956297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151603 - 0.956297i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.691 - 2.12i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.84iT - 3T^{2} \) |
| 7 | \( 1 + 0.145iT - 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 + 5.24iT - 13T^{2} \) |
| 17 | \( 1 - 7.15iT - 17T^{2} \) |
| 23 | \( 1 - 0.622iT - 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 3.77T + 31T^{2} \) |
| 37 | \( 1 + 5.03iT - 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 5.85iT - 47T^{2} \) |
| 53 | \( 1 - 6.97iT - 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 + 8.16T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 + 7.44iT - 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 - 2.17iT - 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 4.98iT - 97T^{2} \) |
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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
−10.67461728148270695704379857903, −10.30045173411237971477761480055, −9.372002458760344938961405143459, −8.259201495191166037945507385241, −7.59105270761020454155853895154, −5.98209622805055950594434696624, −5.53012861508510805039556109674, −4.39883544928620694971361244307, −3.34341541818690546905218718374, −2.59539537682821163918249438841,
0.46195769092392563376062838461, 1.87250313635537423160202761721, 2.70649220817531741865704564532, 4.66919489724046233047418091916, 5.44338573742752083447910001005, 6.50066745918766078389819877394, 7.30091413216339621862414989664, 8.029701919971746771911160665621, 8.835433214466052244505533613055, 9.664074768467531297959307941657
