L(s) = 1 | − 5-s + 2·11-s − 3·17-s + 19-s + 3·23-s + 25-s − 4·29-s + 5·31-s + 10·37-s − 6·41-s + 6·43-s + 8·47-s − 7·49-s − 3·53-s − 2·55-s + 5·61-s + 2·67-s + 2·71-s + 6·73-s + 11·79-s + 9·83-s + 3·85-s − 10·89-s − 95-s + 8·97-s + 12·101-s + 12·103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s − 0.727·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s − 0.742·29-s + 0.898·31-s + 1.64·37-s − 0.937·41-s + 0.914·43-s + 1.16·47-s − 49-s − 0.412·53-s − 0.269·55-s + 0.640·61-s + 0.244·67-s + 0.237·71-s + 0.702·73-s + 1.23·79-s + 0.987·83-s + 0.325·85-s − 1.05·89-s − 0.102·95-s + 0.812·97-s + 1.19·101-s + 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.615692919\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.615692919\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.111444370196918298937918482105, −8.292848490780255118936096255142, −7.54388433483862468747009522560, −6.74532296049614613885841335227, −6.03743477118770305919567612594, −4.95474936882117384170670112667, −4.20645479729920131805946105178, −3.31813773283358528446106337213, −2.22106555349618772060540247053, −0.844954533632952221285850675570,
0.844954533632952221285850675570, 2.22106555349618772060540247053, 3.31813773283358528446106337213, 4.20645479729920131805946105178, 4.95474936882117384170670112667, 6.03743477118770305919567612594, 6.74532296049614613885841335227, 7.54388433483862468747009522560, 8.292848490780255118936096255142, 9.111444370196918298937918482105