L(s) = 1 | + 8.80·2-s + 45.5·4-s + 88.4·5-s − 134.·7-s + 119.·8-s + 779.·10-s − 423.·11-s − 713.·13-s − 1.18e3·14-s − 405.·16-s − 924.·17-s − 64.3·19-s + 4.03e3·20-s − 3.72e3·22-s − 3.70e3·23-s + 4.70e3·25-s − 6.28e3·26-s − 6.11e3·28-s − 4.99e3·29-s + 3.09e3·31-s − 7.39e3·32-s − 8.13e3·34-s − 1.18e4·35-s − 4.97e3·37-s − 566.·38-s + 1.05e4·40-s − 996.·41-s + ⋯ |
L(s) = 1 | + 1.55·2-s + 1.42·4-s + 1.58·5-s − 1.03·7-s + 0.660·8-s + 2.46·10-s − 1.05·11-s − 1.17·13-s − 1.61·14-s − 0.396·16-s − 0.775·17-s − 0.0408·19-s + 2.25·20-s − 1.64·22-s − 1.45·23-s + 1.50·25-s − 1.82·26-s − 1.47·28-s − 1.10·29-s + 0.578·31-s − 1.27·32-s − 1.20·34-s − 1.63·35-s − 0.597·37-s − 0.0636·38-s + 1.04·40-s − 0.0925·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 - 8.80T + 32T^{2} \) |
| 5 | \( 1 - 88.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 134.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 423.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 713.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 924.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 64.3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 996.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.71e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.47e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.14e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.37e5T + 8.58e9T^{2} \) |
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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
−9.893279994075658364169134859169, −8.939452829785856388590654767961, −7.32958812010650247750459866472, −6.43157827174933274357261055959, −5.71742213022146161999262825242, −5.10172127540707433737465093753, −3.90752109352597372831137986630, −2.56525065676672291560114471400, −2.22812316310376252589210869952, 0,
2.22812316310376252589210869952, 2.56525065676672291560114471400, 3.90752109352597372831137986630, 5.10172127540707433737465093753, 5.71742213022146161999262825242, 6.43157827174933274357261055959, 7.32958812010650247750459866472, 8.939452829785856388590654767961, 9.893279994075658364169134859169