L(s) = 1 | + (−1.25 − 0.651i)2-s + (1.15 + 1.63i)4-s − 1.82·5-s + i·7-s + (−0.378 − 2.80i)8-s + (2.29 + 1.18i)10-s − 0.868i·11-s − 0.873i·13-s + (0.651 − 1.25i)14-s + (−1.35 + 3.76i)16-s + 4.00i·17-s − 3.39·19-s + (−2.09 − 2.98i)20-s + (−0.565 + 1.09i)22-s − 3.82·23-s + ⋯ |
L(s) = 1 | + (−0.887 − 0.460i)2-s + (0.575 + 0.817i)4-s − 0.816·5-s + 0.377i·7-s + (−0.133 − 0.990i)8-s + (0.724 + 0.375i)10-s − 0.261i·11-s − 0.242i·13-s + (0.174 − 0.335i)14-s + (−0.337 + 0.941i)16-s + 0.972i·17-s − 0.778·19-s + (−0.469 − 0.667i)20-s + (−0.120 + 0.232i)22-s − 0.798·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0821223 + 0.188682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0821223 + 0.188682i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.25 + 0.651i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 0.868iT - 11T^{2} \) |
| 13 | \( 1 + 0.873iT - 13T^{2} \) |
| 17 | \( 1 - 4.00iT - 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 7.14T + 29T^{2} \) |
| 31 | \( 1 - 4.04iT - 31T^{2} \) |
| 37 | \( 1 + 2.16iT - 37T^{2} \) |
| 41 | \( 1 - 8.89iT - 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 - 9.49T + 53T^{2} \) |
| 59 | \( 1 - 7.42iT - 59T^{2} \) |
| 61 | \( 1 + 2.29iT - 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 3.11iT - 79T^{2} \) |
| 83 | \( 1 - 0.0325iT - 83T^{2} \) |
| 89 | \( 1 - 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−11.18816586814542491415946324243, −10.40501441684596939546472857946, −9.492494429392368101406636337805, −8.401897320611762184837223551489, −8.049494605159421751049153285685, −6.91986105323144092189129379693, −5.86307609122199796659040065168, −4.20517333209737624390303771846, −3.28101052807844608981697629022, −1.82819335555489827517152223412,
0.15599738526607778296338962857, 2.05528666381652873441058480553, 3.77173631131762934444550473085, 4.98425226023319928092124345678, 6.21922512100671290922153397093, 7.21631111378544915148888524065, 7.80105816588502739326616920062, 8.731169331476640093656699322480, 9.650120513590567907237380835166, 10.44046527667460298902836356952