L(s) = 1 | + 2.35·2-s + (−0.813 + 1.52i)3-s + 3.52·4-s − 0.514i·5-s + (−1.91 + 3.59i)6-s + (1.49 + 2.18i)7-s + 3.58·8-s + (−1.67 − 2.48i)9-s − 1.21i·10-s − 2.08·11-s + (−2.86 + 5.39i)12-s + (2.92 − 2.11i)13-s + (3.50 + 5.13i)14-s + (0.787 + 0.418i)15-s + 1.38·16-s + 0.359·17-s + ⋯ |
L(s) = 1 | + 1.66·2-s + (−0.469 + 0.882i)3-s + 1.76·4-s − 0.230i·5-s + (−0.780 + 1.46i)6-s + (0.563 + 0.826i)7-s + 1.26·8-s + (−0.558 − 0.829i)9-s − 0.382i·10-s − 0.629·11-s + (−0.828 + 1.55i)12-s + (0.809 − 0.586i)13-s + (0.936 + 1.37i)14-s + (0.203 + 0.108i)15-s + 0.345·16-s + 0.0871·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.47285 + 0.953673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47285 + 0.953673i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.813 - 1.52i)T \) |
| 7 | \( 1 + (-1.49 - 2.18i)T \) |
| 13 | \( 1 + (-2.92 + 2.11i)T \) |
good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 + 0.514iT - 5T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 17 | \( 1 - 0.359T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 + 1.93iT - 23T^{2} \) |
| 29 | \( 1 + 8.77iT - 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 2.76iT - 37T^{2} \) |
| 41 | \( 1 + 5.37iT - 41T^{2} \) |
| 43 | \( 1 + 0.383T + 43T^{2} \) |
| 47 | \( 1 - 6.90iT - 47T^{2} \) |
| 53 | \( 1 - 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 1.01iT - 59T^{2} \) |
| 61 | \( 1 + 10.7iT - 61T^{2} \) |
| 67 | \( 1 - 3.71iT - 67T^{2} \) |
| 71 | \( 1 + 2.79T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 - 9.78T + 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 - 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
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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
−12.23948478586833413813654027459, −11.16170146950184388507759919134, −10.70253638533220806296192096629, −9.205029315128749511920263876003, −8.124737602101824896339613598584, −6.38845724248896465770067953864, −5.60449925352240257959073315975, −4.88833413758248311547304181127, −3.88203501649542609664817790882, −2.59932799206392336628664533803,
1.83581850296526363995343871704, 3.40543692514491600978836154055, 4.68923145749964571968082266681, 5.59257137928614344090263195052, 6.69722155040114103186959737975, 7.30585887382356666932931829427, 8.594873334724211233397294475396, 10.70890577660989166479766558977, 11.04916314952613846798964629742, 12.01771952055189208562573758446