| L(s) = 1 | − 2·5-s + 11-s − 13-s + 6·17-s − 4·19-s − 8·23-s − 25-s − 10·29-s − 6·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s − 10·53-s − 2·55-s + 12·59-s − 14·61-s + 2·65-s − 12·67-s − 6·73-s − 8·79-s − 12·83-s − 12·85-s − 2·89-s + 8·95-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.85·29-s − 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s − 1.37·53-s − 0.269·55-s + 1.56·59-s − 1.79·61-s + 0.248·65-s − 1.46·67-s − 0.702·73-s − 0.900·79-s − 1.31·83-s − 1.30·85-s − 0.211·89-s + 0.820·95-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.55089494415404, −13.94611577473634, −13.59447227536255, −12.80209956758019, −12.36932743841704, −12.03664656381043, −11.57351673081718, −11.05438634854937, −10.45119329534896, −9.960852863084623, −9.546415398604095, −8.832844058907754, −8.327097993042312, −7.825113546367541, −7.452925778450850, −6.928219111259023, −6.138480423140730, −5.722636538631027, −5.170144075832102, −4.289316870657683, −3.988287946080873, −3.440709354199401, −2.804435372617639, −1.824122490252122, −1.453742543369251, 0, 0,
1.453742543369251, 1.824122490252122, 2.804435372617639, 3.440709354199401, 3.988287946080873, 4.289316870657683, 5.170144075832102, 5.722636538631027, 6.138480423140730, 6.928219111259023, 7.452925778450850, 7.825113546367541, 8.327097993042312, 8.832844058907754, 9.546415398604095, 9.960852863084623, 10.45119329534896, 11.05438634854937, 11.57351673081718, 12.03664656381043, 12.36932743841704, 12.80209956758019, 13.59447227536255, 13.94611577473634, 14.55089494415404
