| L(s) = 1 | − 10.1·3-s + 8.69·5-s + 7·7-s + 75.9·9-s + 11·11-s − 76.3·13-s − 88.2·15-s + 39.7·17-s + 27.9·19-s − 71.0·21-s − 87.2·23-s − 49.3·25-s − 496.·27-s − 38.3·29-s + 186.·31-s − 111.·33-s + 60.8·35-s − 218.·37-s + 774.·39-s + 80.1·41-s + 35.1·43-s + 660.·45-s + 282.·47-s + 49·49-s − 403.·51-s + 145.·53-s + 95.6·55-s + ⋯ |
| L(s) = 1 | − 1.95·3-s + 0.778·5-s + 0.377·7-s + 2.81·9-s + 0.301·11-s − 1.62·13-s − 1.51·15-s + 0.566·17-s + 0.337·19-s − 0.738·21-s − 0.790·23-s − 0.394·25-s − 3.53·27-s − 0.245·29-s + 1.07·31-s − 0.588·33-s + 0.294·35-s − 0.972·37-s + 3.18·39-s + 0.305·41-s + 0.124·43-s + 2.18·45-s + 0.877·47-s + 0.142·49-s − 1.10·51-s + 0.376·53-s + 0.234·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 10.1T + 27T^{2} \) |
| 5 | \( 1 - 8.69T + 125T^{2} \) |
| 13 | \( 1 + 76.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 80.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 35.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 145.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 808.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 794.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 559.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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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
−9.331365291576990630597634661955, −7.78520653904640543621688212872, −7.08985415769147355126255954123, −6.24664216259771538233751562328, −5.51448642916635854182502317376, −4.97986559769831821620795053815, −4.06157890309676685964490331456, −2.22255664627633354396678006536, −1.16552832259182026736239862710, 0,
1.16552832259182026736239862710, 2.22255664627633354396678006536, 4.06157890309676685964490331456, 4.97986559769831821620795053815, 5.51448642916635854182502317376, 6.24664216259771538233751562328, 7.08985415769147355126255954123, 7.78520653904640543621688212872, 9.331365291576990630597634661955
