L(s) = 1 | + (1.09 + 1.09i)3-s − 0.973i·7-s − 0.616i·9-s + (−1.40 + 1.40i)11-s + (−4.60 − 4.60i)13-s + 0.490·17-s + (−4.54 − 4.54i)19-s + (1.06 − 1.06i)21-s − 1.94i·23-s + (3.94 − 3.94i)27-s + (−3.74 − 3.74i)29-s − 4.29·31-s − 3.07·33-s + (4.55 − 4.55i)37-s − 10.0i·39-s + ⋯ |
L(s) = 1 | + (0.630 + 0.630i)3-s − 0.368i·7-s − 0.205i·9-s + (−0.424 + 0.424i)11-s + (−1.27 − 1.27i)13-s + 0.118·17-s + (−1.04 − 1.04i)19-s + (0.231 − 0.231i)21-s − 0.405i·23-s + (0.759 − 0.759i)27-s + (−0.695 − 0.695i)29-s − 0.770·31-s − 0.535·33-s + (0.748 − 0.748i)37-s − 1.60i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147084235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147084235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.09 - 1.09i)T + 3iT^{2} \) |
| 7 | \( 1 + 0.973iT - 7T^{2} \) |
| 11 | \( 1 + (1.40 - 1.40i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.60 + 4.60i)T + 13iT^{2} \) |
| 17 | \( 1 - 0.490T + 17T^{2} \) |
| 19 | \( 1 + (4.54 + 4.54i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.94iT - 23T^{2} \) |
| 29 | \( 1 + (3.74 + 3.74i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + (-4.55 + 4.55i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + (-5.61 + 5.61i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.44 - 8.44i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.01 - 3.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.07 + 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.897iT - 71T^{2} \) |
| 73 | \( 1 - 9.71iT - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + (0.815 + 0.815i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.12iT - 89T^{2} \) |
| 97 | \( 1 + 7.54T + 97T^{2} \) |
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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
−9.342498559951349938929107173296, −8.420528821660405320203653141250, −7.68763534188500083653695692815, −6.94843232239711462900351153310, −5.85043002104788424989702487634, −4.83018908358117304616098694812, −4.18010582820466661204160420766, −3.06472284037687108556718624411, −2.33455078684627712022838956721, −0.37352017714648012541383157869,
1.76721791164901375600833264557, 2.37422246045480885860770404154, 3.53298109421100608316531266846, 4.66440953430037408238648631958, 5.54570159033608021622796597529, 6.55192736696839659856946935885, 7.38928738096102992030742287065, 7.943408464377569714742294166324, 8.813187248106249734902907023440, 9.399820987871871345008217240522