| L(s) = 1 | − 3-s + 7-s − 2·9-s + 11-s − 3·17-s + 7·19-s − 21-s − 23-s + 5·27-s + 3·29-s + 8·31-s − 33-s + 37-s + 11·41-s − 11·43-s − 12·47-s − 6·49-s + 3·51-s + 6·53-s − 7·57-s − 9·59-s − 9·61-s − 2·63-s + 3·67-s + 69-s − 5·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.727·17-s + 1.60·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s + 0.557·29-s + 1.43·31-s − 0.174·33-s + 0.164·37-s + 1.71·41-s − 1.67·43-s − 1.75·47-s − 6/7·49-s + 0.420·51-s + 0.824·53-s − 0.927·57-s − 1.17·59-s − 1.15·61-s − 0.251·63-s + 0.366·67-s + 0.120·69-s − 0.593·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.27475121585743, −14.65284944485612, −14.21056839191461, −13.69306010992014, −13.25834871282003, −12.49024931781212, −11.81122987077702, −11.69783659123267, −11.13447890966217, −10.57249048486201, −9.900603897275665, −9.422977329845649, −8.789995067580814, −8.130524152200807, −7.800342703879520, −6.895315231694712, −6.439596991419801, −5.905971988745966, −5.188737988015026, −4.763897582838969, −4.126262247029942, −3.099821087577820, −2.795822914721381, −1.681114803781715, −0.9842558376838049, 0,
0.9842558376838049, 1.681114803781715, 2.795822914721381, 3.099821087577820, 4.126262247029942, 4.763897582838969, 5.188737988015026, 5.905971988745966, 6.439596991419801, 6.895315231694712, 7.800342703879520, 8.130524152200807, 8.789995067580814, 9.422977329845649, 9.900603897275665, 10.57249048486201, 11.13447890966217, 11.69783659123267, 11.81122987077702, 12.49024931781212, 13.25834871282003, 13.69306010992014, 14.21056839191461, 14.65284944485612, 15.27475121585743
