| L(s) = 1 | + 2·2-s + 8·4-s + 7·5-s + 28·7-s + 40·8-s + 14·10-s − 5·11-s − 28·13-s + 56·14-s + 80·16-s − 21·17-s − 49·19-s + 56·20-s − 10·22-s − 159·23-s + 125·25-s − 56·26-s + 224·28-s − 116·29-s − 147·31-s + 320·32-s − 42·34-s + 196·35-s − 219·37-s − 98·38-s + 280·40-s − 700·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.626·5-s + 1.51·7-s + 1.76·8-s + 0.442·10-s − 0.137·11-s − 0.597·13-s + 1.06·14-s + 5/4·16-s − 0.299·17-s − 0.591·19-s + 0.626·20-s − 0.0969·22-s − 1.44·23-s + 25-s − 0.422·26-s + 1.51·28-s − 0.742·29-s − 0.851·31-s + 1.76·32-s − 0.211·34-s + 0.946·35-s − 0.973·37-s − 0.418·38-s + 1.10·40-s − 2.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.000449091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.000449091\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T - p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T - 76 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T - 1306 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 21 T - 4472 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 49 T - 4458 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 159 T + 13114 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 147 T - 8182 T^{2} + 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 350 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 525 T + 171802 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 303 T - 57068 T^{2} - 303 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 105 T - 194354 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 413 T - 56412 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1113 T + 849752 T^{2} - 1113 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 103 T - 482430 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1092 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 329 T - 596728 T^{2} + 329 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 882 T + p^{3} T^{2} )^{2} \) |
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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
−14.64814336432912702734701116954, −13.95583680330313333603492418147, −13.92538997931754736536572744171, −13.21492657460429433862257146774, −12.46219209517447284072293189274, −12.03613992434757571030065168987, −11.15394390449886121614054168901, −11.09246777481280212376019510064, −10.19612924594517746674523002482, −9.959124515911440903195033045197, −8.558030329400515127888650673747, −8.327961980626200401606484670613, −7.21741002012368570943842159628, −7.11026931262591983094923791019, −5.98008470630566320886347902001, −5.17525991275837360078382387211, −4.73313483146484211874996179363, −3.77791423809809465049800301767, −2.17132250947332270388151761296, −1.75382809136259760563347454696,
1.75382809136259760563347454696, 2.17132250947332270388151761296, 3.77791423809809465049800301767, 4.73313483146484211874996179363, 5.17525991275837360078382387211, 5.98008470630566320886347902001, 7.11026931262591983094923791019, 7.21741002012368570943842159628, 8.327961980626200401606484670613, 8.558030329400515127888650673747, 9.959124515911440903195033045197, 10.19612924594517746674523002482, 11.09246777481280212376019510064, 11.15394390449886121614054168901, 12.03613992434757571030065168987, 12.46219209517447284072293189274, 13.21492657460429433862257146774, 13.92538997931754736536572744171, 13.95583680330313333603492418147, 14.64814336432912702734701116954
