| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 2·13-s − 4·14-s + 16-s − 6·17-s + 4·19-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s − 2·37-s + 4·38-s − 6·41-s − 4·43-s + 9·49-s + 2·52-s − 6·53-s − 4·56-s − 6·58-s + 10·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.277·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s + 1.28·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087590544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087590544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.22973391649383, −13.83436679518390, −13.26256621956431, −13.08749893030217, −12.54862163211068, −11.92786200941264, −11.35417386653792, −11.07561768572633, −10.17177625698024, −9.903911172926840, −9.354330858459932, −8.645945422891212, −8.274674249644647, −7.239090051164330, −7.004395228747958, −6.353040722691544, −6.027707569264893, −5.325053568434956, −4.684243289881667, −4.012007592523062, −3.451957191837842, −2.985272286385384, −2.303186119853407, −1.475718123936882, −0.4328467891184769,
0.4328467891184769, 1.475718123936882, 2.303186119853407, 2.985272286385384, 3.451957191837842, 4.012007592523062, 4.684243289881667, 5.325053568434956, 6.027707569264893, 6.353040722691544, 7.004395228747958, 7.239090051164330, 8.274674249644647, 8.645945422891212, 9.354330858459932, 9.903911172926840, 10.17177625698024, 11.07561768572633, 11.35417386653792, 11.92786200941264, 12.54862163211068, 13.08749893030217, 13.26256621956431, 13.83436679518390, 14.22973391649383
