L(s) = 1 | + 0.544·2-s + 2.32·3-s − 1.70·4-s − 4.20·5-s + 1.26·6-s − 0.904·7-s − 2.01·8-s + 2.40·9-s − 2.28·10-s + 11-s − 3.96·12-s + 5.43·13-s − 0.492·14-s − 9.76·15-s + 2.31·16-s + 0.465·17-s + 1.30·18-s + 1.07·19-s + 7.15·20-s − 2.10·21-s + 0.544·22-s + 5.68·23-s − 4.68·24-s + 12.6·25-s + 2.95·26-s − 1.39·27-s + 1.54·28-s + ⋯ |
L(s) = 1 | + 0.384·2-s + 1.34·3-s − 0.852·4-s − 1.87·5-s + 0.516·6-s − 0.341·7-s − 0.712·8-s + 0.800·9-s − 0.722·10-s + 0.301·11-s − 1.14·12-s + 1.50·13-s − 0.131·14-s − 2.52·15-s + 0.577·16-s + 0.112·17-s + 0.307·18-s + 0.247·19-s + 1.60·20-s − 0.458·21-s + 0.115·22-s + 1.18·23-s − 0.955·24-s + 2.52·25-s + 0.579·26-s − 0.267·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.834200639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834200639\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 - 0.544T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 + 0.904T + 7T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 - 0.465T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 + 5.81T + 53T^{2} \) |
| 59 | \( 1 - 8.84T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−9.133555038857278379272303803694, −8.537773768160939600825149232362, −8.272465307829279115860735311211, −7.35140687797433227307149507642, −6.39269703502655936757985084362, −4.98292901843242558023212788059, −4.09411600993301196082664077260, −3.49521501357764759709804489487, −3.03703370202459158668975279798, −0.891551713247846637800546899132,
0.891551713247846637800546899132, 3.03703370202459158668975279798, 3.49521501357764759709804489487, 4.09411600993301196082664077260, 4.98292901843242558023212788059, 6.39269703502655936757985084362, 7.35140687797433227307149507642, 8.272465307829279115860735311211, 8.537773768160939600825149232362, 9.133555038857278379272303803694