L(s) = 1 | − 0.583·2-s − 3-s − 1.65·4-s + 0.00312·5-s + 0.583·6-s + 2.13·8-s + 9-s − 0.00182·10-s + 4.94·11-s + 1.65·12-s + 13-s − 0.00312·15-s + 2.07·16-s − 5.87·17-s − 0.583·18-s + 5.38·19-s − 0.00519·20-s − 2.88·22-s + 4.23·23-s − 2.13·24-s − 4.99·25-s − 0.583·26-s − 27-s − 8.87·29-s + 0.00182·30-s − 5.36·31-s − 5.48·32-s + ⋯ |
L(s) = 1 | − 0.412·2-s − 0.577·3-s − 0.829·4-s + 0.00139·5-s + 0.238·6-s + 0.755·8-s + 0.333·9-s − 0.000577·10-s + 1.49·11-s + 0.479·12-s + 0.277·13-s − 0.000807·15-s + 0.518·16-s − 1.42·17-s − 0.137·18-s + 1.23·19-s − 0.00116·20-s − 0.615·22-s + 0.882·23-s − 0.435·24-s − 0.999·25-s − 0.114·26-s − 0.192·27-s − 1.64·29-s + 0.000333·30-s − 0.964·31-s − 0.968·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9266438076\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9266438076\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.583T + 2T^{2} \) |
| 5 | \( 1 - 0.00312T + 5T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 1.95T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 - 9.96T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 - 9.08T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.244214363606207568549191635804, −8.733128408957743009985953327039, −7.53693542884532759432696374847, −6.99571038748416685407063315905, −5.93374416084034738464002721713, −5.23293804914317714382743521090, −4.17405424024885681827573459586, −3.68083494562152204600711295298, −1.86393333106623526585966051199, −0.74324660424758292401476420494,
0.74324660424758292401476420494, 1.86393333106623526585966051199, 3.68083494562152204600711295298, 4.17405424024885681827573459586, 5.23293804914317714382743521090, 5.93374416084034738464002721713, 6.99571038748416685407063315905, 7.53693542884532759432696374847, 8.733128408957743009985953327039, 9.244214363606207568549191635804