L(s) = 1 | − 2.62·2-s + 1.21·3-s + 4.91·4-s + 1.03·5-s − 3.19·6-s + 4.97·7-s − 7.66·8-s − 1.52·9-s − 2.72·10-s + 11-s + 5.96·12-s − 4.96·13-s − 13.0·14-s + 1.25·15-s + 10.3·16-s + 17-s + 4.01·18-s + 0.115·19-s + 5.09·20-s + 6.03·21-s − 2.62·22-s + 3.52·23-s − 9.30·24-s − 3.92·25-s + 13.0·26-s − 5.49·27-s + 24.4·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 0.700·3-s + 2.45·4-s + 0.463·5-s − 1.30·6-s + 1.88·7-s − 2.71·8-s − 0.508·9-s − 0.861·10-s + 0.301·11-s + 1.72·12-s − 1.37·13-s − 3.49·14-s + 0.324·15-s + 2.58·16-s + 0.242·17-s + 0.946·18-s + 0.0265·19-s + 1.13·20-s + 1.31·21-s − 0.560·22-s + 0.734·23-s − 1.89·24-s − 0.785·25-s + 2.56·26-s − 1.05·27-s + 4.62·28-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
\(L(1)\) |
\(\approx\) |
\(1.527648323\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.527648323\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 1.21T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 4.97T + 7T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 19 | \( 1 - 0.115T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 + 0.835T + 37T^{2} \) |
| 41 | \( 1 - 9.76T + 41T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + 7.89T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 5.98T + 89T^{2} \) |
| 97 | \( 1 + 9.81T + 97T^{2} \) |
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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
−7.959475547130706227913893711491, −7.61948385912995486868558126105, −6.84549136874932266046487023977, −5.96653081008715833784855849461, −5.14246780311166678814131246735, −4.32343071554495503572784825636, −2.78708694187010716726626990954, −2.45242282315895685671431973595, −1.63309479075753078277671461738, −0.812862622323073198613536583003,
0.812862622323073198613536583003, 1.63309479075753078277671461738, 2.45242282315895685671431973595, 2.78708694187010716726626990954, 4.32343071554495503572784825636, 5.14246780311166678814131246735, 5.96653081008715833784855849461, 6.84549136874932266046487023977, 7.61948385912995486868558126105, 7.959475547130706227913893711491