| 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
| \(L(2)\) |
\(\approx\) |
\(0.09319840905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09319840905\) |
| \(L(\frac{5}{2})\) |
|
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
−11.22718953650145456524634811916, −10.69848568910595134278503104771, −10.08022753709341111813784559010, −9.502930871250312603934364001010, −9.153412735762435813365293280761, −8.452752075024644117360338190241, −8.300456059768788335934309772723, −7.61080674785986353736222703592, −7.32415913510733469585724220316, −6.88696586871679177847606564364, −6.24897192492978602376124384222, −6.06798497284611620469518502712, −5.06257030267533155752074775259, −4.35687027431981637604899661711, −4.33357641705958660798518579158, −3.26690197134692543060934417574, −2.63600098156871553705481678495, −2.42399043678284600912043526879, −1.26982408475306296995562172691, −0.10648037426824505281491924198,
0.10648037426824505281491924198, 1.26982408475306296995562172691, 2.42399043678284600912043526879, 2.63600098156871553705481678495, 3.26690197134692543060934417574, 4.33357641705958660798518579158, 4.35687027431981637604899661711, 5.06257030267533155752074775259, 6.06798497284611620469518502712, 6.24897192492978602376124384222, 6.88696586871679177847606564364, 7.32415913510733469585724220316, 7.61080674785986353736222703592, 8.300456059768788335934309772723, 8.452752075024644117360338190241, 9.153412735762435813365293280761, 9.502930871250312603934364001010, 10.08022753709341111813784559010, 10.69848568910595134278503104771, 11.22718953650145456524634811916
