| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 2·11-s + 4·13-s + 15-s + 6·17-s + 4·19-s + 21-s + 23-s + 25-s + 27-s + 2·29-s − 2·31-s − 2·33-s + 35-s + 8·37-s + 4·39-s + 6·41-s + 45-s − 8·47-s + 49-s + 6·51-s − 8·53-s − 2·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 1.31·37-s + 0.640·39-s + 0.937·41-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.09·53-s − 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.623103656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.623103656\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.65139285274460, −14.34027223035037, −13.83957302682529, −13.30651965394276, −12.85011337350040, −12.37525697995057, −11.54895919638848, −11.21260772043791, −10.55357524284974, −9.951630195265633, −9.599762260979139, −8.974879728735646, −8.331386369005613, −7.865939480644679, −7.493317852101456, −6.685085789430362, −6.004583094644462, −5.513125654635802, −4.939518752980024, −4.191663948134905, −3.405970577014662, −3.012650027122852, −2.214971204763470, −1.373838621492835, −0.8459628250249836,
0.8459628250249836, 1.373838621492835, 2.214971204763470, 3.012650027122852, 3.405970577014662, 4.191663948134905, 4.939518752980024, 5.513125654635802, 6.004583094644462, 6.685085789430362, 7.493317852101456, 7.865939480644679, 8.331386369005613, 8.974879728735646, 9.599762260979139, 9.951630195265633, 10.55357524284974, 11.21260772043791, 11.54895919638848, 12.37525697995057, 12.85011337350040, 13.30651965394276, 13.83957302682529, 14.34027223035037, 14.65139285274460
