| L(s) = 1 | + 5·5-s − 30·7-s + 71·11-s − 35·13-s − 38·19-s + 5·23-s − 25·25-s − 155·29-s − 88·31-s − 150·35-s + 380·37-s + 142·41-s + 155·43-s + 455·47-s + 121·49-s + 275·53-s + 355·55-s + 873·59-s + 445·61-s − 175·65-s + 645·67-s + 1.71e3·71-s − 990·73-s − 2.13e3·77-s − 1.27e3·79-s + 90·83-s + 888·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.61·7-s + 1.94·11-s − 0.746·13-s − 0.458·19-s + 0.0453·23-s − 1/5·25-s − 0.992·29-s − 0.509·31-s − 0.724·35-s + 1.68·37-s + 0.540·41-s + 0.549·43-s + 1.41·47-s + 0.352·49-s + 0.712·53-s + 0.870·55-s + 1.92·59-s + 0.934·61-s − 0.333·65-s + 1.17·67-s + 2.86·71-s − 1.58·73-s − 3.15·77-s − 1.81·79-s + 0.119·83-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.511874879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511874879\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - p T + 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 71 T + 3716 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 35 T + 3702 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9529 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T - 4378 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 380 T + 136878 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 142 T + 135458 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 155 T + 52086 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 455 T + 255038 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 275 T + 191846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 645 T + 674834 T^{2} - 645 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1712 T + 1445258 T^{2} - 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1274 T + 1391022 T^{2} + 1274 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 90 T + 1038382 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 888 T + 1554274 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.898282919824676467819061210740, −9.876920074749917127980533239077, −9.369822017367636944263315997193, −9.367710768538648362207267906348, −8.594270085373175938500606386800, −8.362679277712691515174423425778, −7.31134187279783502398446733001, −7.29899088293097704781948736841, −6.68936825838579083153604166412, −6.37022001322190435601788386528, −5.73059494679876164812070175265, −5.71503613927961295749636720782, −4.70844037546864279938284075004, −4.14488322680177391411288563237, −3.73926525562612193182535782394, −3.33038469008129037410598407720, −2.38698577873984063923039018663, −2.16226772878340744705854340515, −1.06294406286461980016720156497, −0.51345303785227750599809352763,
0.51345303785227750599809352763, 1.06294406286461980016720156497, 2.16226772878340744705854340515, 2.38698577873984063923039018663, 3.33038469008129037410598407720, 3.73926525562612193182535782394, 4.14488322680177391411288563237, 4.70844037546864279938284075004, 5.71503613927961295749636720782, 5.73059494679876164812070175265, 6.37022001322190435601788386528, 6.68936825838579083153604166412, 7.29899088293097704781948736841, 7.31134187279783502398446733001, 8.362679277712691515174423425778, 8.594270085373175938500606386800, 9.367710768538648362207267906348, 9.369822017367636944263315997193, 9.876920074749917127980533239077, 9.898282919824676467819061210740
