| L(s) = 1 | − 3·3-s − 2.84·5-s − 31.6·7-s + 9·9-s − 11·11-s − 5.15·13-s + 8.54·15-s + 121.·17-s + 34.8·19-s + 95.0·21-s − 116.·23-s − 116.·25-s − 27·27-s + 69.4·29-s − 140.·31-s + 33·33-s + 90.3·35-s + 420.·37-s + 15.4·39-s − 322.·41-s + 321.·43-s − 25.6·45-s + 231.·47-s + 661.·49-s − 365.·51-s − 4.91·53-s + 31.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.254·5-s − 1.71·7-s + 0.333·9-s − 0.301·11-s − 0.109·13-s + 0.147·15-s + 1.73·17-s + 0.420·19-s + 0.988·21-s − 1.05·23-s − 0.935·25-s − 0.192·27-s + 0.444·29-s − 0.814·31-s + 0.174·33-s + 0.436·35-s + 1.86·37-s + 0.0634·39-s − 1.22·41-s + 1.13·43-s − 0.0849·45-s + 0.718·47-s + 1.92·49-s − 1.00·51-s − 0.0127·53-s + 0.0768·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 2.84T + 125T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 13 | \( 1 + 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−8.205539235379976929871056113425, −7.51164920471636100458862320444, −6.74383831625972180987990187596, −5.86061046059353646431171744937, −5.48618737980421678762834421003, −4.09059139512440551722830023729, −3.46519715384380952153481499954, −2.48643864884181134921350768366, −0.942119045351771133279059287173, 0,
0.942119045351771133279059287173, 2.48643864884181134921350768366, 3.46519715384380952153481499954, 4.09059139512440551722830023729, 5.48618737980421678762834421003, 5.86061046059353646431171744937, 6.74383831625972180987990187596, 7.51164920471636100458862320444, 8.205539235379976929871056113425
