| L(s) = 1 | − 2·7-s + 11-s + 13-s + 17-s − 4·19-s + 6·23-s − 5·25-s + 8·29-s − 2·31-s + 4·37-s + 6·41-s + 4·43-s + 8·47-s − 3·49-s + 6·53-s + 12·59-s − 12·61-s + 12·67-s + 14·71-s − 10·73-s − 2·77-s + 6·79-s + 4·83-s − 6·89-s − 2·91-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s + 1.25·23-s − 25-s + 1.48·29-s − 0.359·31-s + 0.657·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s − 1.53·61-s + 1.46·67-s + 1.66·71-s − 1.17·73-s − 0.227·77-s + 0.675·79-s + 0.439·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.993607415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.993607415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.64431691073523, −12.14459903191679, −11.68677091458001, −11.09428374223956, −10.78253592663581, −10.32283909799507, −9.675306707267518, −9.503107418621533, −8.882846672239769, −8.494697495954014, −8.018153305103798, −7.401984621156366, −6.967882158477696, −6.500471167631814, −6.054937534053544, −5.657673996029946, −5.001407082236806, −4.443725466324191, −3.929023979101903, −3.546082616764704, −2.781894693067628, −2.484428445679798, −1.752884076591893, −0.9073311452186930, −0.5581606368726265,
0.5581606368726265, 0.9073311452186930, 1.752884076591893, 2.484428445679798, 2.781894693067628, 3.546082616764704, 3.929023979101903, 4.443725466324191, 5.001407082236806, 5.657673996029946, 6.054937534053544, 6.500471167631814, 6.967882158477696, 7.401984621156366, 8.018153305103798, 8.494697495954014, 8.882846672239769, 9.503107418621533, 9.675306707267518, 10.32283909799507, 10.78253592663581, 11.09428374223956, 11.68677091458001, 12.14459903191679, 12.64431691073523
