L(s) = 1 | − 0.259·3-s − 1.96·5-s + 4.76·7-s − 2.93·9-s − 3.31·11-s + 0.493·13-s + 0.509·15-s − 3.02·17-s − 1.99·19-s − 1.23·21-s − 3.65·23-s − 1.14·25-s + 1.53·27-s + 5.29·29-s + 8.88·31-s + 0.859·33-s − 9.35·35-s + 6.88·37-s − 0.127·39-s − 3.94·41-s − 10.3·43-s + 5.75·45-s + 12.7·47-s + 15.6·49-s + 0.785·51-s + 5.75·53-s + 6.50·55-s + ⋯ |
L(s) = 1 | − 0.149·3-s − 0.878·5-s + 1.80·7-s − 0.977·9-s − 0.998·11-s + 0.136·13-s + 0.131·15-s − 0.734·17-s − 0.457·19-s − 0.269·21-s − 0.761·23-s − 0.228·25-s + 0.296·27-s + 0.983·29-s + 1.59·31-s + 0.149·33-s − 1.58·35-s + 1.13·37-s − 0.0204·39-s − 0.615·41-s − 1.58·43-s + 0.858·45-s + 1.86·47-s + 2.24·49-s + 0.109·51-s + 0.790·53-s + 0.877·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.353462505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353462505\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + 0.259T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 - 4.76T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 - 0.493T + 13T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 8.88T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 5.75T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 8.70T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.420763532149976900588170946107, −7.937809165355605386334783102158, −7.17630162114432478198656773295, −6.12954716429041824351954163706, −5.36459406205643971763069471367, −4.62298581930946546717264084744, −4.11284793793116013737869822164, −2.81364816474457901127193573354, −2.07062856396230468854062154361, −0.65705871577796517167448261819,
0.65705871577796517167448261819, 2.07062856396230468854062154361, 2.81364816474457901127193573354, 4.11284793793116013737869822164, 4.62298581930946546717264084744, 5.36459406205643971763069471367, 6.12954716429041824351954163706, 7.17630162114432478198656773295, 7.937809165355605386334783102158, 8.420763532149976900588170946107