| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 11-s − 13-s + 2·14-s + 16-s − 7·19-s + 22-s + 9·23-s − 5·25-s + 26-s − 2·28-s + 9·29-s + 7·31-s − 32-s − 2·37-s + 7·38-s − 12·41-s − 4·43-s − 44-s − 9·46-s − 9·47-s − 3·49-s + 5·50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.301·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.60·19-s + 0.213·22-s + 1.87·23-s − 25-s + 0.196·26-s − 0.377·28-s + 1.67·29-s + 1.25·31-s − 0.176·32-s − 0.328·37-s + 1.13·38-s − 1.87·41-s − 0.609·43-s − 0.150·44-s − 1.32·46-s − 1.31·47-s − 3/7·49-s + 0.707·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7024941838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7024941838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.56275311304620, −13.65949869120987, −13.34536563318324, −12.89257782751426, −12.18464317959733, −11.89187418143615, −11.23674199968858, −10.57569067681930, −10.24213394559889, −9.836765117993983, −9.187960675833606, −8.593750859587542, −8.300154284163326, −7.666630288910560, −6.889921790686941, −6.466340769767147, −6.295202834327166, −5.128635952963987, −4.885869105433512, −4.001119134023472, −3.216084317422962, −2.785891755120401, −2.065406372067849, −1.255087571288870, −0.3347682300421471,
0.3347682300421471, 1.255087571288870, 2.065406372067849, 2.785891755120401, 3.216084317422962, 4.001119134023472, 4.885869105433512, 5.128635952963987, 6.295202834327166, 6.466340769767147, 6.889921790686941, 7.666630288910560, 8.300154284163326, 8.593750859587542, 9.187960675833606, 9.836765117993983, 10.24213394559889, 10.57569067681930, 11.23674199968858, 11.89187418143615, 12.18464317959733, 12.89257782751426, 13.34536563318324, 13.65949869120987, 14.56275311304620
