| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s − 13-s + 14-s + 16-s − 2·19-s + 20-s + 22-s − 8·23-s + 25-s − 26-s + 28-s − 2·29-s + 32-s + 35-s + 4·37-s − 2·38-s + 40-s + 2·41-s + 10·43-s + 44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s − 0.371·29-s + 0.176·32-s + 0.169·35-s + 0.657·37-s − 0.324·38-s + 0.158·40-s + 0.312·41-s + 1.52·43-s + 0.150·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.25054000150013, −13.60002547145716, −13.20748743928667, −12.64121347434570, −12.20327170992584, −11.75179148170904, −11.20938737925643, −10.73885209310561, −10.20555347346551, −9.676625708684983, −9.216019005294259, −8.548364536709113, −7.928404889944217, −7.590130435557963, −6.887946913421899, −6.274148771867199, −5.940182723787789, −5.392718541530911, −4.705792906526546, −4.242694951351832, −3.749288950160661, −2.966377609673922, −2.298102580800932, −1.849636334690448, −1.089037767844692, 0,
1.089037767844692, 1.849636334690448, 2.298102580800932, 2.966377609673922, 3.749288950160661, 4.242694951351832, 4.705792906526546, 5.392718541530911, 5.940182723787789, 6.274148771867199, 6.887946913421899, 7.590130435557963, 7.928404889944217, 8.548364536709113, 9.216019005294259, 9.676625708684983, 10.20555347346551, 10.73885209310561, 11.20938737925643, 11.75179148170904, 12.20327170992584, 12.64121347434570, 13.20748743928667, 13.60002547145716, 14.25054000150013
