L(s) = 1 | − 3-s − 3.57·5-s + 7-s + 9-s + 4.81·11-s − 3.91·13-s + 3.57·15-s − 7.21·17-s + 0.335·19-s − 21-s + 23-s + 7.79·25-s − 27-s + 7.11·29-s + 1.21·31-s − 4.81·33-s − 3.57·35-s + 1.55·37-s + 3.91·39-s − 0.0422·41-s − 8.12·43-s − 3.57·45-s − 4.10·47-s + 49-s + 7.21·51-s + 6.35·53-s − 17.2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.59·5-s + 0.377·7-s + 0.333·9-s + 1.45·11-s − 1.08·13-s + 0.923·15-s − 1.75·17-s + 0.0769·19-s − 0.218·21-s + 0.208·23-s + 1.55·25-s − 0.192·27-s + 1.32·29-s + 0.218·31-s − 0.838·33-s − 0.604·35-s + 0.256·37-s + 0.626·39-s − 0.00659·41-s − 1.23·43-s − 0.533·45-s − 0.598·47-s + 0.142·49-s + 1.01·51-s + 0.872·53-s − 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.57T + 5T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 - 0.335T + 19T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.0422T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.19T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.32208037861349891265726702591, −6.87746898480833242884900889391, −6.41257146065624493497161467518, −5.19973263868615367513059241316, −4.46394387604214902351691251387, −4.22982228330512844970581926760, −3.29232301597384254593855214515, −2.21188199843812964952850075081, −1.00308738090636194998451174759, 0,
1.00308738090636194998451174759, 2.21188199843812964952850075081, 3.29232301597384254593855214515, 4.22982228330512844970581926760, 4.46394387604214902351691251387, 5.19973263868615367513059241316, 6.41257146065624493497161467518, 6.87746898480833242884900889391, 7.32208037861349891265726702591