| L(s) = 1 | + 2.40i·3-s + (1.83 − 1.28i)5-s + 0.706i·7-s − 2.79·9-s − 0.747·11-s + 1.29i·13-s + (3.09 + 4.40i)15-s + 5.50i·17-s + 2.44·19-s − 1.70·21-s − i·23-s + (1.70 − 4.70i)25-s + 0.483i·27-s − 5.72·29-s − 7.52·31-s + ⋯ |
| L(s) = 1 | + 1.39i·3-s + (0.818 − 0.574i)5-s + 0.267i·7-s − 0.933·9-s − 0.225·11-s + 0.358i·13-s + (0.798 + 1.13i)15-s + 1.33i·17-s + 0.561·19-s − 0.371·21-s − 0.208i·23-s + (0.340 − 0.940i)25-s + 0.0930i·27-s − 1.06·29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.778010797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778010797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.83 + 1.28i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.40iT - 3T^{2} \) |
| 7 | \( 1 - 0.706iT - 7T^{2} \) |
| 11 | \( 1 + 0.747T + 11T^{2} \) |
| 13 | \( 1 - 1.29iT - 13T^{2} \) |
| 17 | \( 1 - 5.50iT - 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 29 | \( 1 + 5.72T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 5.07iT - 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.34iT - 43T^{2} \) |
| 47 | \( 1 - 7.90iT - 47T^{2} \) |
| 53 | \( 1 - 5.84iT - 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.57T + 61T^{2} \) |
| 67 | \( 1 - 5.25iT - 67T^{2} \) |
| 71 | \( 1 + 2.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.55T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 1.50iT - 97T^{2} \) |
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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
−9.527779190838386259318446460293, −8.992494278058320480770321803370, −8.278065603966166418616594337519, −7.15512149383742875481040194339, −5.83828941517399836083308428636, −5.59074543650840547832592385818, −4.51365545197555594245959122036, −3.94103318616754138541135936276, −2.75474079422005080693226161935, −1.52482652232490779202594214871,
0.65438480873931064870453112590, 1.91305495953829818186867156487, 2.62269364485462266410545050575, 3.73990982730062571433652153136, 5.36019271202944019830766408326, 5.73410997131736588997955363065, 6.95392261519260550973779354748, 7.17166259807410657685913352797, 7.87715931988137355999723678574, 9.038202109769047832017966680785
