| L(s) = 1 | − 2-s + 8·4-s + 16·5-s − 23·8-s − 16·10-s − 8·11-s + 56·13-s + 23·16-s + 54·17-s + 110·19-s + 128·20-s + 8·22-s + 48·23-s + 125·25-s − 56·26-s + 220·29-s − 12·31-s − 184·32-s − 54·34-s + 246·37-s − 110·38-s − 368·40-s − 364·41-s + 256·43-s − 64·44-s − 48·46-s + 324·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 4-s + 1.43·5-s − 1.01·8-s − 0.505·10-s − 0.219·11-s + 1.19·13-s + 0.359·16-s + 0.770·17-s + 1.32·19-s + 1.43·20-s + 0.0775·22-s + 0.435·23-s + 25-s − 0.422·26-s + 1.40·29-s − 0.0695·31-s − 1.01·32-s − 0.272·34-s + 1.09·37-s − 0.469·38-s − 1.45·40-s − 1.38·41-s + 0.907·43-s − 0.219·44-s − 0.153·46-s + 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.091741443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.091741443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 16 T + 131 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T - 1267 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 54 T - 1997 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 110 T + 5241 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 48 T - 9863 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T - 29647 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 324 T + 1153 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 162 T - 122633 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 810 T + 450721 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 p T + 3 p^{2} T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 244 T - 241227 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 702 T + 103787 T^{2} - 702 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 440 T - 299439 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1302 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 730 T - 172069 T^{2} - 730 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
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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
−10.71144173652892578800200083046, −10.59585924695174955441166854341, −10.00050785191616953959729341498, −9.497396529554490287377856646897, −9.331409932415387736550207966272, −8.757515657563788254236679152446, −8.142523448754037315593696388968, −7.86614210966335957494909599128, −6.98879511510992930375663025225, −6.79257997461267644919280994153, −6.12607085621097607827704306220, −5.98370478788842618888121749082, −5.21760689363701894683483349736, −5.04390503295634203479544022675, −3.74904758289954947892421270786, −3.37965081365274067006023698990, −2.43530115610110639598646489259, −2.37038076601518511088725144966, −1.21673811027979800978683288484, −0.895727967893125989130151281851,
0.895727967893125989130151281851, 1.21673811027979800978683288484, 2.37038076601518511088725144966, 2.43530115610110639598646489259, 3.37965081365274067006023698990, 3.74904758289954947892421270786, 5.04390503295634203479544022675, 5.21760689363701894683483349736, 5.98370478788842618888121749082, 6.12607085621097607827704306220, 6.79257997461267644919280994153, 6.98879511510992930375663025225, 7.86614210966335957494909599128, 8.142523448754037315593696388968, 8.757515657563788254236679152446, 9.331409932415387736550207966272, 9.497396529554490287377856646897, 10.00050785191616953959729341498, 10.59585924695174955441166854341, 10.71144173652892578800200083046
