| L(s) = 1 | + 2.39i·3-s + 3.99·5-s − i·7-s − 2.72·9-s + 0.655i·11-s − 1.32i·13-s + 9.55i·15-s + 1.55i·17-s + 3.21i·19-s + 2.39·21-s − 1.09·23-s + 10.9·25-s + 0.649i·27-s + 4.14i·29-s − 3.62·31-s + ⋯ |
| L(s) = 1 | + 1.38i·3-s + 1.78·5-s − 0.377i·7-s − 0.909·9-s + 0.197i·11-s − 0.366i·13-s + 2.46i·15-s + 0.376i·17-s + 0.738i·19-s + 0.522·21-s − 0.227·23-s + 2.19·25-s + 0.124i·27-s + 0.769i·29-s − 0.650·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.250391727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250391727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 41 | \( 1 + (-6.37 - 0.647i)T \) |
| good | 3 | \( 1 - 2.39iT - 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 11 | \( 1 - 0.655iT - 11T^{2} \) |
| 13 | \( 1 + 1.32iT - 13T^{2} \) |
| 17 | \( 1 - 1.55iT - 17T^{2} \) |
| 19 | \( 1 - 3.21iT - 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 4.14iT - 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 - 2.13iT - 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 8.90T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 0.603iT - 67T^{2} \) |
| 71 | \( 1 + 5.43iT - 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.77iT - 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 2.88iT - 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−9.893208402568332506934057893250, −9.491450163011612241329802430337, −8.744993267190305908919918514858, −7.55097239398819915487469970066, −6.31553346802213274588893409026, −5.66986128792103373684426740405, −4.89588236308472070231121976441, −3.94833376383608321554508165451, −2.83908719725874019027132174548, −1.58281570360802164499167929284,
1.08079239197544101896157192332, 2.13676369801936873152442171111, 2.71666980715267998635146107648, 4.61366840397067679629402567174, 5.88949042829303500901775174505, 6.06186054374468198379122433473, 7.04135899273419380903858631542, 7.79113956963584070486903590771, 9.009825984049446641792661180728, 9.351579234368623936172443065602
