| L(s) = 1 | + 2·2-s + 4·4-s − 14·7-s + 8·8-s + 6·11-s − 68·13-s − 28·14-s + 16·16-s − 78·17-s + 44·19-s + 12·22-s − 120·23-s − 136·26-s − 56·28-s + 126·29-s − 244·31-s + 32·32-s − 156·34-s + 304·37-s + 88·38-s − 480·41-s − 104·43-s + 24·44-s − 240·46-s − 600·47-s − 147·49-s − 272·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.164·11-s − 1.45·13-s − 0.534·14-s + 1/4·16-s − 1.11·17-s + 0.531·19-s + 0.116·22-s − 1.08·23-s − 1.02·26-s − 0.377·28-s + 0.806·29-s − 1.41·31-s + 0.176·32-s − 0.786·34-s + 1.35·37-s + 0.375·38-s − 1.82·41-s − 0.368·43-s + 0.0822·44-s − 0.769·46-s − 1.86·47-s − 3/7·49-s − 0.725·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 + 244 T + p^{3} T^{2} \) |
| 37 | \( 1 - 304 T + p^{3} T^{2} \) |
| 41 | \( 1 + 480 T + p^{3} T^{2} \) |
| 43 | \( 1 + 104 T + p^{3} T^{2} \) |
| 47 | \( 1 + 600 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 534 T + p^{3} T^{2} \) |
| 61 | \( 1 - 362 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 972 T + p^{3} T^{2} \) |
| 73 | \( 1 + 470 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1244 T + p^{3} T^{2} \) |
| 83 | \( 1 + 396 T + p^{3} T^{2} \) |
| 89 | \( 1 + 972 T + p^{3} T^{2} \) |
| 97 | \( 1 - 46 T + p^{3} T^{2} \) |
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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
−10.13049925464263967133121220028, −9.581098889231599572095801033639, −8.314915698362690280686589401539, −7.16117608992018946034823285100, −6.48892652833559398649459278934, −5.34277530744560152544143363481, −4.36871482013104047618058952777, −3.19877124876594439541589003356, −2.06732181658361691458486863163, 0,
2.06732181658361691458486863163, 3.19877124876594439541589003356, 4.36871482013104047618058952777, 5.34277530744560152544143363481, 6.48892652833559398649459278934, 7.16117608992018946034823285100, 8.314915698362690280686589401539, 9.581098889231599572095801033639, 10.13049925464263967133121220028
