L(s) = 1 | − 2i·3-s + (2 − i)5-s + 3i·7-s − 9-s + 6i·13-s + (−2 − 4i)15-s − 7i·17-s − 4·19-s + 6·21-s − i·23-s + (3 − 4i)25-s − 4i·27-s + 9·29-s + 3·31-s + (3 + 6i)35-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + (0.894 − 0.447i)5-s + 1.13i·7-s − 0.333·9-s + 1.66i·13-s + (−0.516 − 1.03i)15-s − 1.69i·17-s − 0.917·19-s + 1.30·21-s − 0.208i·23-s + (0.600 − 0.800i)25-s − 0.769i·27-s + 1.67·29-s + 0.538·31-s + (0.507 + 1.01i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.133489578\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.133489578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 7iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 - 13T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.106880328259528099367033204138, −8.454109571540342954702242546988, −7.41921580403427661255451879341, −6.55897312714781837698762306884, −6.20508359996827198648627932013, −5.13324573187851431949735853493, −4.36357284262363655520652507468, −2.44472970356902299281897618415, −2.24498697136300666590928618800, −0.947255169095409542204540926973,
1.16587171903868198005473538032, 2.69073401660752880830005209203, 3.63720928975287075157645829245, 4.36214722805439604445009313855, 5.27769580530086317560682466396, 6.14661255766429408260920904217, 6.86338147016201561933453114650, 8.014136237444527536420658788535, 8.583204717326241163457665832685, 9.891853816327565069451934964429