| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s + 12-s + 6·13-s − 15-s − 16-s + 6·17-s − 18-s − 4·19-s − 20-s − 23-s − 3·24-s + 25-s − 6·26-s − 27-s + 29-s + 30-s + 4·31-s − 5·32-s − 6·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.208·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.185·29-s + 0.182·30-s + 0.718·31-s − 0.883·32-s − 1.02·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.88644428130964, −16.64227392130719, −15.98454716216685, −15.37719515724761, −14.53424262090718, −14.01054207083566, −13.38015187196821, −13.02620300029173, −12.24992405011988, −11.62422815269201, −10.86988005922789, −10.34971449187829, −10.01515990262597, −9.225012542492432, −8.612667083519166, −8.138961488895020, −7.465487784512962, −6.498317784933029, −6.067242633235919, −5.319905779295503, −4.621152616997111, −3.852862596770265, −3.083669522176013, −1.620927367156824, −1.215602962736103, 0,
1.215602962736103, 1.620927367156824, 3.083669522176013, 3.852862596770265, 4.621152616997111, 5.319905779295503, 6.067242633235919, 6.498317784933029, 7.465487784512962, 8.138961488895020, 8.612667083519166, 9.225012542492432, 10.01515990262597, 10.34971449187829, 10.86988005922789, 11.62422815269201, 12.24992405011988, 13.02620300029173, 13.38015187196821, 14.01054207083566, 14.53424262090718, 15.37719515724761, 15.98454716216685, 16.64227392130719, 16.88644428130964
