L(s) = 1 | − 1.88·2-s − 1.81·3-s + 1.53·4-s + 2.13·5-s + 3.40·6-s + 1.99·7-s + 0.875·8-s + 0.284·9-s − 4.02·10-s − 11-s − 2.78·12-s + 4.25·13-s − 3.75·14-s − 3.87·15-s − 4.71·16-s + 17-s − 0.534·18-s − 0.707·19-s + 3.28·20-s − 3.62·21-s + 1.88·22-s − 3.06·23-s − 1.58·24-s − 0.426·25-s − 7.99·26-s + 4.92·27-s + 3.06·28-s + ⋯ |
L(s) = 1 | − 1.32·2-s − 1.04·3-s + 0.767·4-s + 0.956·5-s + 1.39·6-s + 0.755·7-s + 0.309·8-s + 0.0947·9-s − 1.27·10-s − 0.301·11-s − 0.802·12-s + 1.17·13-s − 1.00·14-s − 1.00·15-s − 1.17·16-s + 0.242·17-s − 0.125·18-s − 0.162·19-s + 0.733·20-s − 0.790·21-s + 0.400·22-s − 0.639·23-s − 0.323·24-s − 0.0852·25-s − 1.56·26-s + 0.947·27-s + 0.579·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6788636245\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6788636245\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 19 | \( 1 + 0.707T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 5.06T + 31T^{2} \) |
| 37 | \( 1 - 0.886T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 47 | \( 1 + 0.190T + 47T^{2} \) |
| 53 | \( 1 - 0.911T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 3.89T + 61T^{2} \) |
| 67 | \( 1 - 0.454T + 67T^{2} \) |
| 71 | \( 1 + 7.59T + 71T^{2} \) |
| 73 | \( 1 + 1.67T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.988201699947967511518146676915, −7.22785354314192252071843797634, −6.44948371330626029533393124561, −5.80879315936383684217148861264, −5.31090897558607048611115367394, −4.51553463913863125435126342567, −3.44299713490334917732409548901, −2.02099558464868064934583892825, −1.60523593121241780221083064845, −0.54764745802591033574501958514,
0.54764745802591033574501958514, 1.60523593121241780221083064845, 2.02099558464868064934583892825, 3.44299713490334917732409548901, 4.51553463913863125435126342567, 5.31090897558607048611115367394, 5.80879315936383684217148861264, 6.44948371330626029533393124561, 7.22785354314192252071843797634, 7.988201699947967511518146676915