| L(s) = 1 | − 0.546·2-s − 3-s − 1.70·4-s + 0.546·6-s + 7-s + 2.02·8-s + 9-s − 11-s + 1.70·12-s + 0.568·13-s − 0.546·14-s + 2.29·16-s − 2.70·17-s − 0.546·18-s + 6.62·19-s − 21-s + 0.546·22-s − 8.93·23-s − 2.02·24-s − 0.310·26-s − 27-s − 1.70·28-s + 5.27·29-s + 9.66·31-s − 5.29·32-s + 33-s + 1.47·34-s + ⋯ |
| L(s) = 1 | − 0.386·2-s − 0.577·3-s − 0.850·4-s + 0.223·6-s + 0.377·7-s + 0.714·8-s + 0.333·9-s − 0.301·11-s + 0.491·12-s + 0.157·13-s − 0.146·14-s + 0.574·16-s − 0.655·17-s − 0.128·18-s + 1.51·19-s − 0.218·21-s + 0.116·22-s − 1.86·23-s − 0.412·24-s − 0.0609·26-s − 0.192·27-s − 0.321·28-s + 0.978·29-s + 1.73·31-s − 0.936·32-s + 0.174·33-s + 0.253·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9371931432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9371931432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.546T + 2T^{2} \) |
| 13 | \( 1 - 0.568T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 + 8.93T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.0520T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 6.36T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.260731414695086006446136190455, −7.53815025590403700932486760969, −6.73316583920013599717823638605, −5.85472809228459640972098591725, −5.22373245359172516437959550231, −4.51131554937494080369075549063, −3.91554808833790533205079724864, −2.75574755716487798576331052246, −1.54010501741424244806542317877, −0.60043861093834606515294138406,
0.60043861093834606515294138406, 1.54010501741424244806542317877, 2.75574755716487798576331052246, 3.91554808833790533205079724864, 4.51131554937494080369075549063, 5.22373245359172516437959550231, 5.85472809228459640972098591725, 6.73316583920013599717823638605, 7.53815025590403700932486760969, 8.260731414695086006446136190455
