| L(s) = 1 | + 1.12·2-s + 3-s − 0.742·4-s + 3.71·5-s + 1.12·6-s + 4.45·7-s − 3.07·8-s + 9-s + 4.16·10-s − 0.158·11-s − 0.742·12-s − 2.56·13-s + 5.00·14-s + 3.71·15-s − 1.96·16-s − 17-s + 1.12·18-s + 1.14·19-s − 2.75·20-s + 4.45·21-s − 0.177·22-s + 5.04·23-s − 3.07·24-s + 8.76·25-s − 2.87·26-s + 27-s − 3.31·28-s + ⋯ |
| L(s) = 1 | + 0.792·2-s + 0.577·3-s − 0.371·4-s + 1.65·5-s + 0.457·6-s + 1.68·7-s − 1.08·8-s + 0.333·9-s + 1.31·10-s − 0.0476·11-s − 0.214·12-s − 0.710·13-s + 1.33·14-s + 0.958·15-s − 0.490·16-s − 0.242·17-s + 0.264·18-s + 0.263·19-s − 0.616·20-s + 0.973·21-s − 0.0378·22-s + 1.05·23-s − 0.627·24-s + 1.75·25-s − 0.563·26-s + 0.192·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.986203713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.986203713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 11 | \( 1 + 0.158T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 + 0.702T + 31T^{2} \) |
| 37 | \( 1 - 2.30T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 - 8.78T + 53T^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 3.48T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 2.52T + 97T^{2} \) |
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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
−8.574318043376793132894764116155, −7.76059827538447376679458765104, −6.89528085698796133004745968819, −5.94916005966421145879052929350, −5.16354119360768628249317069734, −4.94775297543424894564161649820, −4.01467211494910081849859465625, −2.81882873196381448357802231244, −2.16241478320285299016392022496, −1.24585058476198793220378553504,
1.24585058476198793220378553504, 2.16241478320285299016392022496, 2.81882873196381448357802231244, 4.01467211494910081849859465625, 4.94775297543424894564161649820, 5.16354119360768628249317069734, 5.94916005966421145879052929350, 6.89528085698796133004745968819, 7.76059827538447376679458765104, 8.574318043376793132894764116155
