| L(s) = 1 | + 8·5-s + 24·7-s + 2·11-s − 32·13-s + 56·17-s − 184·19-s − 92·23-s + 95·25-s − 336·29-s + 376·31-s + 192·35-s − 348·37-s + 312·41-s − 80·43-s + 228·47-s + 43·49-s + 152·53-s + 16·55-s + 680·59-s + 112·61-s − 256·65-s − 352·67-s − 1.81e3·71-s − 574·73-s + 48·77-s − 1.36e3·79-s − 782·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s + 1.29·7-s + 0.0548·11-s − 0.682·13-s + 0.798·17-s − 2.22·19-s − 0.834·23-s + 0.759·25-s − 2.15·29-s + 2.17·31-s + 0.927·35-s − 1.54·37-s + 1.18·41-s − 0.283·43-s + 0.707·47-s + 0.125·49-s + 0.393·53-s + 0.0392·55-s + 1.50·59-s + 0.235·61-s − 0.488·65-s − 0.641·67-s − 3.03·71-s − 0.920·73-s + 0.0710·77-s − 1.93·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.632950029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632950029\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T - 31 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 24 T + 533 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 32 T - 102 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 56 T + 5858 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 184 T + 20994 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 p T + 21698 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 336 T + 75814 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 376 T + 87501 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 348 T + 126830 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 312 T + 132478 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 80 T + 102402 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 228 T + 201634 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 152 T + 253337 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 680 T + 355286 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 112 T + 152970 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 352 T + 289170 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1816 T + 1497518 T^{2} + 1816 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 287 T + p^{3} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 1360 T + 1144350 T^{2} + 1360 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 782 T + 992327 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 240 T + 1328110 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1834899 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.853676269187949345645125035656, −8.852598961083197197528789846046, −8.399694191618968559856412956072, −8.093538476940265831923523915758, −7.39875492841859669811307863933, −7.39572265751219176598361302431, −6.77603529730494763106951721783, −6.30052751824040150246827721832, −5.72231247347491442967717350859, −5.70481139395128206556376314217, −5.07878392384342030377394022638, −4.60850188533501466780050058982, −4.08959380001062514371271192783, −4.06808914870257712703490613288, −2.98438216033985609552491641886, −2.66623405372861977018391255910, −2.00385119534232641666693405488, −1.71251861922386186568511081970, −1.20137973860444151305595288673, −0.26302693884667498888440145457,
0.26302693884667498888440145457, 1.20137973860444151305595288673, 1.71251861922386186568511081970, 2.00385119534232641666693405488, 2.66623405372861977018391255910, 2.98438216033985609552491641886, 4.06808914870257712703490613288, 4.08959380001062514371271192783, 4.60850188533501466780050058982, 5.07878392384342030377394022638, 5.70481139395128206556376314217, 5.72231247347491442967717350859, 6.30052751824040150246827721832, 6.77603529730494763106951721783, 7.39572265751219176598361302431, 7.39875492841859669811307863933, 8.093538476940265831923523915758, 8.399694191618968559856412956072, 8.852598961083197197528789846046, 8.853676269187949345645125035656
