L(s) = 1 | − 2·5-s + 4·7-s + 11-s + 4·13-s + 17-s − 8·19-s − 25-s − 10·31-s − 8·35-s − 8·37-s + 10·41-s − 8·43-s + 10·47-s + 9·49-s − 12·53-s − 2·55-s + 8·59-s + 2·61-s − 8·65-s + 4·67-s − 4·71-s + 10·73-s + 4·77-s − 12·79-s − 4·83-s − 2·85-s + 6·89-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.301·11-s + 1.10·13-s + 0.242·17-s − 1.83·19-s − 1/5·25-s − 1.79·31-s − 1.35·35-s − 1.31·37-s + 1.56·41-s − 1.21·43-s + 1.45·47-s + 9/7·49-s − 1.64·53-s − 0.269·55-s + 1.04·59-s + 0.256·61-s − 0.992·65-s + 0.488·67-s − 0.474·71-s + 1.17·73-s + 0.455·77-s − 1.35·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.23093587722528, −13.36024897145235, −12.95866380370763, −12.32377987488672, −12.01747994844592, −11.29803013955386, −11.00679768396496, −10.83516901644211, −10.16217602126119, −9.305200915612768, −8.854358471293256, −8.300226954641359, −8.159974208277645, −7.512292879574425, −6.982343622003507, −6.436053219274789, −5.654253163977611, −5.367647422170864, −4.489575722239879, −4.134446963169419, −3.769823751485002, −3.031123050963685, −1.918854241038231, −1.821846306173694, −0.8834935160505292, 0,
0.8834935160505292, 1.821846306173694, 1.918854241038231, 3.031123050963685, 3.769823751485002, 4.134446963169419, 4.489575722239879, 5.367647422170864, 5.654253163977611, 6.436053219274789, 6.982343622003507, 7.512292879574425, 8.159974208277645, 8.300226954641359, 8.854358471293256, 9.305200915612768, 10.16217602126119, 10.83516901644211, 11.00679768396496, 11.29803013955386, 12.01747994844592, 12.32377987488672, 12.95866380370763, 13.36024897145235, 14.23093587722528