| L(s) = 1 | + (−0.671 + 1.24i)2-s + (−1.09 − 1.67i)4-s − 4.68·7-s + (2.81 − 0.244i)8-s − 2.29i·11-s + 4.97i·13-s + (3.14 − 5.83i)14-s + (−1.58 + 3.67i)16-s − 2.97·17-s + 2.68i·19-s + (2.85 + 1.53i)22-s + 2.68·23-s + (−6.19 − 3.34i)26-s + (5.14 + 7.83i)28-s − 2i·29-s + ⋯ |
| L(s) = 1 | + (−0.474 + 0.880i)2-s + (−0.549 − 0.835i)4-s − 1.77·7-s + (0.996 − 0.0864i)8-s − 0.691i·11-s + 1.38i·13-s + (0.840 − 1.55i)14-s + (−0.396 + 0.917i)16-s − 0.722·17-s + 0.616i·19-s + (0.608 + 0.328i)22-s + 0.560·23-s + (−1.21 − 0.655i)26-s + (0.972 + 1.48i)28-s − 0.371i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6988581199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6988581199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.671 - 1.24i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + 2.29iT - 11T^{2} \) |
| 13 | \( 1 - 4.97iT - 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 - 2.68iT - 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 4.39iT - 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.37iT - 43T^{2} \) |
| 47 | \( 1 - 7.27T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 4.58iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 - 3.95T + 97T^{2} \) |
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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
−9.045690978866952239457215387354, −8.844761573587171324821165331443, −7.46586516678719022721698128332, −6.93333165290188897371131584082, −6.16554600486580354687516715374, −5.66506106151958854032916660480, −4.31260100468863441349918369131, −3.56799846344778704449239776755, −2.14860982376489530472917910572, −0.45001769388990119508104840004,
0.78331585753042317096143118586, 2.46381231146871155679198501522, 3.09370184393510576662341524532, 3.97559211195128537200151651355, 5.04086208578218352807201386837, 6.14647806078355211117935525180, 7.07243383472208918058813963618, 7.70292529110054875509673596568, 8.853193602159574728907917102255, 9.355951502592591340507429450728
