L(s) = 1 | − 2·4-s + 2·5-s + 2·7-s − 11-s + 5·13-s + 4·16-s − 3·17-s − 4·20-s + 6·23-s − 25-s − 4·28-s − 10·29-s + 10·31-s + 4·35-s − 10·41-s − 6·43-s + 2·44-s − 2·47-s − 3·49-s − 10·52-s + 5·53-s − 2·55-s − 5·59-s − 8·64-s + 10·65-s + 10·67-s + 6·68-s + ⋯ |
L(s) = 1 | − 4-s + 0.894·5-s + 0.755·7-s − 0.301·11-s + 1.38·13-s + 16-s − 0.727·17-s − 0.894·20-s + 1.25·23-s − 1/5·25-s − 0.755·28-s − 1.85·29-s + 1.79·31-s + 0.676·35-s − 1.56·41-s − 0.914·43-s + 0.301·44-s − 0.291·47-s − 3/7·49-s − 1.38·52-s + 0.686·53-s − 0.269·55-s − 0.650·59-s − 64-s + 1.24·65-s + 1.22·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.20000186506632, −14.61735518601448, −13.91428042642458, −13.58809721849215, −13.22381794424949, −12.90937008064049, −12.01840872316523, −11.35105545378201, −11.02011109431793, −10.30848739950714, −9.858214407594245, −9.262615814370497, −8.715658243207720, −8.365140018920674, −7.809644016288665, −6.963133385854783, −6.301251266320398, −5.793177744122787, −5.030109457901862, −4.873000405281182, −3.924241000999192, −3.439646382175132, −2.520121546672903, −1.634658197562532, −1.165494150200080, 0,
1.165494150200080, 1.634658197562532, 2.520121546672903, 3.439646382175132, 3.924241000999192, 4.873000405281182, 5.030109457901862, 5.793177744122787, 6.301251266320398, 6.963133385854783, 7.809644016288665, 8.365140018920674, 8.715658243207720, 9.262615814370497, 9.858214407594245, 10.30848739950714, 11.02011109431793, 11.35105545378201, 12.01840872316523, 12.90937008064049, 13.22381794424949, 13.58809721849215, 13.91428042642458, 14.61735518601448, 15.20000186506632