L(s) = 1 | + 1.76i·3-s + (−0.432 + 2.19i)5-s + i·7-s − 0.103·9-s + 0.626·11-s + 5.49i·13-s + (−3.86 − 0.761i)15-s + 0.896i·17-s + 6.38·19-s − 1.76·21-s + 3.72i·23-s + (−4.62 − 1.89i)25-s + 5.10i·27-s − 7.87·29-s + 7.52·31-s + ⋯ |
L(s) = 1 | + 1.01i·3-s + (−0.193 + 0.981i)5-s + 0.377i·7-s − 0.0343·9-s + 0.188·11-s + 1.52i·13-s + (−0.997 − 0.196i)15-s + 0.217i·17-s + 1.46·19-s − 0.384·21-s + 0.777i·23-s + (−0.925 − 0.379i)25-s + 0.982i·27-s − 1.46·29-s + 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.671714834\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.671714834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.432 - 2.19i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 1.76iT - 3T^{2} \) |
| 11 | \( 1 - 0.626T + 11T^{2} \) |
| 13 | \( 1 - 5.49iT - 13T^{2} \) |
| 17 | \( 1 - 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 + 6.77iT - 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.72iT - 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−9.551005088950665402624959127195, −8.934546331687011028659111326592, −7.69104042477904539213787915656, −7.15808557227403169982629071852, −6.23698440604603865904396956571, −5.42328740734099249821448923412, −4.37997368707955189165282506806, −3.78121504769123861144496794781, −2.90955795261554342058630492544, −1.69937272077775759384195679956,
0.64255595511844002602397299692, 1.29342535865369396081798939881, 2.63535842161519751676928039204, 3.72886529994956842613722420228, 4.76367226253344250100429414234, 5.52900567277069603578072012337, 6.36387605061267676880912095775, 7.38916244582176228321618119458, 7.79207366707342687931220006538, 8.423951186964024090491522978376