| L(s) = 1 | + 2-s + 1.18·3-s + 4-s − 0.675·5-s + 1.18·6-s + 3.42·7-s + 8-s − 1.58·9-s − 0.675·10-s + 3.44·11-s + 1.18·12-s − 5.49·13-s + 3.42·14-s − 0.801·15-s + 16-s + 4.89·17-s − 1.58·18-s − 2.61·19-s − 0.675·20-s + 4.06·21-s + 3.44·22-s − 23-s + 1.18·24-s − 4.54·25-s − 5.49·26-s − 5.45·27-s + 3.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.685·3-s + 0.5·4-s − 0.301·5-s + 0.484·6-s + 1.29·7-s + 0.353·8-s − 0.529·9-s − 0.213·10-s + 1.03·11-s + 0.342·12-s − 1.52·13-s + 0.915·14-s − 0.207·15-s + 0.250·16-s + 1.18·17-s − 0.374·18-s − 0.599·19-s − 0.150·20-s + 0.887·21-s + 0.734·22-s − 0.208·23-s + 0.242·24-s − 0.908·25-s − 1.07·26-s − 1.04·27-s + 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.564938426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.564938426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 + 0.675T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 - 6.03T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 - 7.85T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 0.960T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 - 2.25T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.904720887488471292078837407967, −7.62495054068559137611747589065, −6.63520250878086425658658096226, −5.83520107590606105206746372026, −5.05725400819452470938719883164, −4.38293811972079436335484634077, −3.77615519542751470681222522627, −2.71135465606860720633261922806, −2.18964853524254141903164390646, −1.01510267636811992592502111562,
1.01510267636811992592502111562, 2.18964853524254141903164390646, 2.71135465606860720633261922806, 3.77615519542751470681222522627, 4.38293811972079436335484634077, 5.05725400819452470938719883164, 5.83520107590606105206746372026, 6.63520250878086425658658096226, 7.62495054068559137611747589065, 7.904720887488471292078837407967
