| L(s) = 1 | − 3-s − 3·5-s + 9-s − 2·13-s + 3·15-s − 17-s + 4·19-s + 6·23-s + 4·25-s − 27-s − 8·31-s − 2·37-s + 2·39-s − 6·41-s + 9·43-s − 3·45-s + 9·47-s + 51-s + 8·53-s − 4·57-s + 5·59-s − 10·61-s + 6·65-s + 13·67-s − 6·69-s + 12·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.554·13-s + 0.774·15-s − 0.242·17-s + 0.917·19-s + 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.43·31-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.37·43-s − 0.447·45-s + 1.31·47-s + 0.140·51-s + 1.09·53-s − 0.529·57-s + 0.650·59-s − 1.28·61-s + 0.744·65-s + 1.58·67-s − 0.722·69-s + 1.42·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.70288371332971, −12.45178643655279, −12.05467367577223, −11.57601285992542, −11.15149986776441, −10.81398217998020, −10.44096992068939, −9.552804278224178, −9.431617577001901, −8.764552318867356, −8.273127387580856, −7.737235020370197, −7.242416766568997, −7.054043503378772, −6.553900020232594, −5.622121365240257, −5.419363750026516, −4.852737126602719, −4.346292919351952, −3.683190139822747, −3.519474036168707, −2.666086353946875, −2.141071129325009, −1.182495206397874, −0.6870923459348033, 0,
0.6870923459348033, 1.182495206397874, 2.141071129325009, 2.666086353946875, 3.519474036168707, 3.683190139822747, 4.346292919351952, 4.852737126602719, 5.419363750026516, 5.622121365240257, 6.553900020232594, 7.054043503378772, 7.242416766568997, 7.737235020370197, 8.273127387580856, 8.764552318867356, 9.431617577001901, 9.552804278224178, 10.44096992068939, 10.81398217998020, 11.15149986776441, 11.57601285992542, 12.05467367577223, 12.45178643655279, 12.70288371332971
