L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 4·13-s − 17-s − 2·19-s + 2·21-s − 2·23-s − 27-s − 2·29-s + 8·31-s + 33-s − 10·37-s + 4·39-s − 6·41-s + 6·43-s − 4·47-s − 3·49-s + 51-s + 2·57-s − 14·59-s + 6·61-s − 2·63-s + 2·69-s − 14·71-s + 16·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 0.417·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.914·43-s − 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.264·57-s − 1.82·59-s + 0.768·61-s − 0.251·63-s + 0.240·69-s − 1.66·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.94814850714913, −12.79453770043789, −12.23755267521396, −11.79356902559668, −11.50337714581515, −10.70296539099600, −10.38076692459385, −10.01418567775759, −9.571773887146509, −8.995406843574746, −8.546801912673995, −7.853276880572302, −7.496326808797307, −6.887574256921094, −6.469329694517943, −6.094653609702207, −5.464415032125473, −4.830849466915348, −4.659492281059096, −3.844974928658370, −3.315822319696329, −2.721882307594243, −2.118467340528266, −1.519781834648802, −0.5324471964420342, 0,
0.5324471964420342, 1.519781834648802, 2.118467340528266, 2.721882307594243, 3.315822319696329, 3.844974928658370, 4.659492281059096, 4.830849466915348, 5.464415032125473, 6.094653609702207, 6.469329694517943, 6.887574256921094, 7.496326808797307, 7.853276880572302, 8.546801912673995, 8.995406843574746, 9.571773887146509, 10.01418567775759, 10.38076692459385, 10.70296539099600, 11.50337714581515, 11.79356902559668, 12.23755267521396, 12.79453770043789, 12.94814850714913