L(s) = 1 | − 2·4-s + 5-s − 4·7-s + 2·11-s + 13-s + 4·16-s − 4·19-s − 2·20-s − 4·23-s + 25-s + 8·28-s − 29-s + 3·31-s − 4·35-s + 6·37-s + 12·41-s − 8·43-s − 4·44-s + 6·47-s + 9·49-s − 2·52-s + 11·53-s + 2·55-s + 7·59-s − 8·64-s + 65-s − 9·67-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s + 0.603·11-s + 0.277·13-s + 16-s − 0.917·19-s − 0.447·20-s − 0.834·23-s + 1/5·25-s + 1.51·28-s − 0.185·29-s + 0.538·31-s − 0.676·35-s + 0.986·37-s + 1.87·41-s − 1.21·43-s − 0.603·44-s + 0.875·47-s + 9/7·49-s − 0.277·52-s + 1.51·53-s + 0.269·55-s + 0.911·59-s − 64-s + 0.124·65-s − 1.09·67-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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.42739835424768, −13.11191539525623, −12.61461429388327, −12.16769203472153, −11.76071820458522, −10.95034748364046, −10.41375060568030, −10.08664629093524, −9.595859797877853, −9.130283656646124, −8.950988118388611, −8.214126574033610, −7.804046526375941, −7.034375942926067, −6.505315770454471, −6.116235832732795, −5.735193929710713, −5.114995990995863, −4.307065354770345, −3.973453492439407, −3.599219772795402, −2.737562180793610, −2.370428775477206, −1.355704653568563, −0.7175603907739108, 0,
0.7175603907739108, 1.355704653568563, 2.370428775477206, 2.737562180793610, 3.599219772795402, 3.973453492439407, 4.307065354770345, 5.114995990995863, 5.735193929710713, 6.116235832732795, 6.505315770454471, 7.034375942926067, 7.804046526375941, 8.214126574033610, 8.950988118388611, 9.130283656646124, 9.595859797877853, 10.08664629093524, 10.41375060568030, 10.95034748364046, 11.76071820458522, 12.16769203472153, 12.61461429388327, 13.11191539525623, 13.42739835424768