| L(s) = 1 | + 1.91·3-s + 5-s + 3.32i·7-s + 0.669·9-s − 5.61·11-s + 3.09i·13-s + 1.91·15-s + (−3.99 − 1.03i)17-s + 0.142i·19-s + 6.37i·21-s − 1.97i·23-s + 25-s − 4.46·27-s + 1.39·29-s + 5.08i·31-s + ⋯ |
| L(s) = 1 | + 1.10·3-s + 0.447·5-s + 1.25i·7-s + 0.223·9-s − 1.69·11-s + 0.859i·13-s + 0.494·15-s + (−0.967 − 0.252i)17-s + 0.0327i·19-s + 1.39i·21-s − 0.411i·23-s + 0.200·25-s − 0.859·27-s + 0.259·29-s + 0.912i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434751895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434751895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + (3.99 + 1.03i)T \) |
| good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 7 | \( 1 - 3.32iT - 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 - 3.09iT - 13T^{2} \) |
| 19 | \( 1 - 0.142iT - 19T^{2} \) |
| 23 | \( 1 + 1.97iT - 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 - 5.08iT - 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 4.91iT - 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 + 0.833iT - 53T^{2} \) |
| 59 | \( 1 + 1.78iT - 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 8.46iT - 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 4.67iT - 97T^{2} \) |
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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
−8.979861363659086940345977344919, −8.508902836466546831141442208064, −7.86341605981366161887824168462, −6.90542350427433610803337772915, −6.00825955088637750492941039600, −5.23510550347759613462382733673, −4.45806201172780549998479920136, −3.07151191551264184425874159459, −2.55494217742510082559672150168, −1.91360007145924604569983712198,
0.34539252527599426788114602756, 1.93411224933812061394219147888, 2.77992627337007722771650562634, 3.51233520471401187527911906055, 4.51109884705117566101742492953, 5.36448474886055840697305712710, 6.28541294253808709079802879106, 7.35260292281405920159900667656, 7.83685731155563814111369236202, 8.372569478503532291186243190139
