| L(s) = 1 | + 0.388·2-s + 3.09·3-s − 7.84·4-s + 16.8·5-s + 1.20·6-s + 26.1·7-s − 6.15·8-s − 17.3·9-s + 6.56·10-s − 11·11-s − 24.3·12-s − 13·13-s + 10.1·14-s + 52.3·15-s + 60.4·16-s + 80.2·17-s − 6.75·18-s + 97.0·19-s − 132.·20-s + 81.0·21-s − 4.27·22-s + 173.·23-s − 19.0·24-s + 160.·25-s − 5.04·26-s − 137.·27-s − 205.·28-s + ⋯ |
| L(s) = 1 | + 0.137·2-s + 0.596·3-s − 0.981·4-s + 1.51·5-s + 0.0819·6-s + 1.41·7-s − 0.271·8-s − 0.644·9-s + 0.207·10-s − 0.301·11-s − 0.585·12-s − 0.277·13-s + 0.193·14-s + 0.901·15-s + 0.943·16-s + 1.14·17-s − 0.0884·18-s + 1.17·19-s − 1.48·20-s + 0.842·21-s − 0.0413·22-s + 1.57·23-s − 0.162·24-s + 1.28·25-s − 0.0380·26-s − 0.980·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.392474913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392474913\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 0.388T + 8T^{2} \) |
| 3 | \( 1 - 3.09T + 27T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 17 | \( 1 - 80.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 40.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 31.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 313.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 27.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 305.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 998.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 548.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 596.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 370.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 928.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−13.04384142847190315543621163106, −11.68096178368377988215431117410, −10.36764002261953547639248786532, −9.383062283877443885297431296326, −8.657059573548045268170708483744, −7.56388083861520171056872805870, −5.50077929358564335685800233306, −5.13368781673347296956644867363, −3.14049277280726439502337956496, −1.51858572269427002679256576497,
1.51858572269427002679256576497, 3.14049277280726439502337956496, 5.13368781673347296956644867363, 5.50077929358564335685800233306, 7.56388083861520171056872805870, 8.657059573548045268170708483744, 9.383062283877443885297431296326, 10.36764002261953547639248786532, 11.68096178368377988215431117410, 13.04384142847190315543621163106
