| L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s + 0.618·6-s + 3·7-s + 2.23·8-s + 9-s + 1.61·12-s + 4.85·13-s − 1.85·14-s + 1.85·16-s − 1.47·17-s − 0.618·18-s + 7.23·19-s − 3·21-s − 5.38·23-s − 2.23·24-s − 3.00·26-s − 27-s − 4.85·28-s − 1.38·29-s − 2.14·31-s − 5.61·32-s + 0.909·34-s − 1.61·36-s − 2.14·37-s − 4.47·38-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.252·6-s + 1.13·7-s + 0.790·8-s + 0.333·9-s + 0.467·12-s + 1.34·13-s − 0.495·14-s + 0.463·16-s − 0.357·17-s − 0.145·18-s + 1.66·19-s − 0.654·21-s − 1.12·23-s − 0.456·24-s − 0.588·26-s − 0.192·27-s − 0.917·28-s − 0.256·29-s − 0.385·31-s − 0.993·32-s + 0.156·34-s − 0.269·36-s − 0.352·37-s − 0.725·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 0.145T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.61654525827892912681544626671, −6.76849024713045363299505953477, −5.81992966520887133200303347149, −5.36342494109812180066380740425, −4.63183694104525777612342143144, −4.02146284315080136480662303728, −3.20919893245594078758774762000, −1.65295468296667353948395928053, −1.27159492024378143510262470889, 0,
1.27159492024378143510262470889, 1.65295468296667353948395928053, 3.20919893245594078758774762000, 4.02146284315080136480662303728, 4.63183694104525777612342143144, 5.36342494109812180066380740425, 5.81992966520887133200303347149, 6.76849024713045363299505953477, 7.61654525827892912681544626671
