L(s) = 1 | − 7-s − 11-s − 2·13-s − 3·17-s + 7·19-s + 3·23-s − 6·29-s + 4·31-s − 5·37-s − 3·41-s − 4·43-s + 9·47-s − 6·49-s + 6·53-s − 3·59-s − 10·61-s − 10·67-s − 15·71-s − 2·73-s + 77-s + 79-s − 6·89-s + 2·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.60·19-s + 0.625·23-s − 1.11·29-s + 0.718·31-s − 0.821·37-s − 0.468·41-s − 0.609·43-s + 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.390·59-s − 1.28·61-s − 1.22·67-s − 1.78·71-s − 0.234·73-s + 0.113·77-s + 0.112·79-s − 0.635·89-s + 0.209·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.423201436\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423201436\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.90542398729354, −14.20182392921739, −13.69635978001825, −13.27798697758614, −12.82154353085732, −12.03650739437149, −11.80298793662118, −11.17449249481793, −10.47753855548647, −10.13013342398868, −9.415686761230500, −9.066395490434469, −8.470090485545607, −7.647931418473273, −7.281959105236809, −6.817422916525793, −5.959335557267299, −5.547305905331257, −4.804088117818703, −4.382003615174701, −3.302356894390142, −3.109590753844871, −2.191620618281945, −1.436686589646066, −0.4370314560870123,
0.4370314560870123, 1.436686589646066, 2.191620618281945, 3.109590753844871, 3.302356894390142, 4.382003615174701, 4.804088117818703, 5.547305905331257, 5.959335557267299, 6.817422916525793, 7.281959105236809, 7.647931418473273, 8.470090485545607, 9.066395490434469, 9.415686761230500, 10.13013342398868, 10.47753855548647, 11.17449249481793, 11.80298793662118, 12.03650739437149, 12.82154353085732, 13.27798697758614, 13.69635978001825, 14.20182392921739, 14.90542398729354