| L(s) = 1 | + 2-s − 4-s − 5-s + 4·7-s − 3·8-s − 10-s − 4·11-s + 13-s + 4·14-s − 16-s − 4·19-s + 20-s − 4·22-s + 25-s + 26-s − 4·28-s − 6·29-s + 5·32-s − 4·35-s + 10·37-s − 4·38-s + 3·40-s − 10·41-s − 12·43-s + 4·44-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s − 1.06·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 1.11·29-s + 0.883·32-s − 0.676·35-s + 1.64·37-s − 0.648·38-s + 0.474·40-s − 1.56·41-s − 1.82·43-s + 0.603·44-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.42064461228633, −13.04547383899357, −12.69471492716161, −12.00441200320322, −11.61497079000457, −11.22633169122145, −10.75082015040224, −10.19000882214641, −9.764735664974266, −8.867256061618409, −8.686119215218819, −8.096399635841471, −7.834403669551042, −7.254394625169727, −6.553906658252047, −5.872051640378739, −5.496992464798187, −4.839033831556908, −4.705746979164256, −4.080103943391224, −3.532283246758768, −2.923082822128112, −2.202813286591000, −1.685140443622661, −0.7317922496157011, 0,
0.7317922496157011, 1.685140443622661, 2.202813286591000, 2.923082822128112, 3.532283246758768, 4.080103943391224, 4.705746979164256, 4.839033831556908, 5.496992464798187, 5.872051640378739, 6.553906658252047, 7.254394625169727, 7.834403669551042, 8.096399635841471, 8.686119215218819, 8.867256061618409, 9.764735664974266, 10.19000882214641, 10.75082015040224, 11.22633169122145, 11.61497079000457, 12.00441200320322, 12.69471492716161, 13.04547383899357, 13.42064461228633
