L(s) = 1 | + (0.599 + 1.28i)2-s + (0.468 + 1.66i)3-s + (−1.28 + 1.53i)4-s + (−1.85 + 1.59i)6-s + 0.936i·7-s + (−2.73 − 0.719i)8-s + (−2.56 + 1.56i)9-s + 4.27·11-s + (−3.16 − 1.41i)12-s − 3.12·13-s + (−1.19 + 0.561i)14-s + (−0.719 − 3.93i)16-s − 2i·17-s + (−3.53 − 2.34i)18-s + 4.27i·19-s + ⋯ |
L(s) = 1 | + (0.424 + 0.905i)2-s + (0.270 + 0.962i)3-s + (−0.640 + 0.768i)4-s + (−0.757 + 0.653i)6-s + 0.353i·7-s + (−0.967 − 0.254i)8-s + (−0.853 + 0.520i)9-s + 1.28·11-s + (−0.912 − 0.408i)12-s − 0.866·13-s + (−0.320 + 0.150i)14-s + (−0.179 − 0.983i)16-s − 0.485i·17-s + (−0.833 − 0.552i)18-s + 0.979i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.322564 + 1.50865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.322564 + 1.50865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.599 - 1.28i)T \) |
| 3 | \( 1 + (-0.468 - 1.66i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.936iT - 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4.27iT - 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 0.936T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 5.20iT - 67T^{2} \) |
| 71 | \( 1 + 6.67T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 9.06iT - 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−12.12764167978280654515811177536, −11.40038525077060349283111084078, −9.900665557533711259419073194819, −9.241695430907347409749615747277, −8.411590393392659239737778043016, −7.27150643198848724146627923729, −6.11534148820643516121561489640, −5.05193064724565745811613278584, −4.13857923000341105035196113773, −2.95329091601968759064014084546,
1.07712393993001698090376670091, 2.52318802995770322804562207815, 3.78031368100233113485143075041, 5.12080329401952024716331123853, 6.46110575297323141700411624909, 7.29302970223211834061978374573, 8.806639932437364156285032282824, 9.340433563600901625255114221810, 10.69059956106067348883884219570, 11.52424961294030993524189531197