L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 2·13-s + 2·19-s − 2·21-s − 27-s − 8·31-s − 33-s + 2·37-s − 2·39-s − 2·43-s − 3·49-s + 6·53-s − 2·57-s − 12·59-s − 2·61-s + 2·63-s + 4·67-s − 2·73-s + 2·77-s + 10·79-s + 81-s + 12·83-s − 6·89-s + 4·91-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 0.436·21-s − 0.192·27-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 0.304·43-s − 3/7·49-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.234·73-s + 0.227·77-s + 1.12·79-s + 1/9·81-s + 1.31·83-s − 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.81145488418416, −14.13503304934052, −13.75341110908278, −13.17945817761252, −12.57042704352489, −12.16422709794272, −11.51387566909507, −11.18234914139282, −10.75735862049049, −10.17077633527507, −9.499448168559340, −9.048299867950435, −8.458823500866258, −7.788280923500972, −7.432142476673968, −6.694513255927098, −6.206175356123614, −5.559569811860627, −5.096521408307136, −4.505264304888195, −3.827798936620099, −3.295435268389742, −2.326950170444910, −1.600164487303859, −1.041587016101660, 0,
1.041587016101660, 1.600164487303859, 2.326950170444910, 3.295435268389742, 3.827798936620099, 4.505264304888195, 5.096521408307136, 5.559569811860627, 6.206175356123614, 6.694513255927098, 7.432142476673968, 7.788280923500972, 8.458823500866258, 9.048299867950435, 9.499448168559340, 10.17077633527507, 10.75735862049049, 11.18234914139282, 11.51387566909507, 12.16422709794272, 12.57042704352489, 13.17945817761252, 13.75341110908278, 14.13503304934052, 14.81145488418416