| L(s) = 1 | − 4·4-s − 120·11-s + 16·16-s − 22·19-s + 192·29-s + 40·31-s − 384·41-s + 480·44-s + 637·49-s + 312·59-s + 166·61-s − 64·64-s − 432·71-s + 88·76-s + 1.05e3·79-s + 72·89-s − 2.49e3·101-s + 3.88e3·109-s − 768·116-s + 8.13e3·121-s − 160·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.28·11-s + 1/4·16-s − 0.265·19-s + 1.22·29-s + 0.231·31-s − 1.46·41-s + 1.64·44-s + 13/7·49-s + 0.688·59-s + 0.348·61-s − 1/8·64-s − 0.722·71-s + 0.132·76-s + 1.50·79-s + 0.0857·89-s − 2.45·101-s + 3.41·109-s − 0.614·116-s + 6.11·121-s − 0.115·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.054258444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054258444\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 60 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1847 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1838 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 72745 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 192 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 79130 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 166030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 168154 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 156 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 83 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 599317 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 216 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 p^{2} T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 529 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 128810 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1459321 T^{2} + p^{6} T^{4} \) |
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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
−9.529517495869652014777545323572, −8.876294685338878260001849519074, −8.431336248087487117452259236269, −8.392355685733742628052314985348, −7.87274201515483449585722434233, −7.41233094473798580137022766887, −7.23897220155182911351228848108, −6.51588379207177079750209058628, −6.08885116421139115631609832648, −5.48798976383435345079509547902, −5.24840843080219393777388302118, −4.90558428126005564635363790841, −4.49177206350173556172620669691, −3.83474277693139782937637541223, −3.26619671876105743918501083144, −2.60668542296021568850806672292, −2.54678111081278821672367778482, −1.78117217895741255629094341070, −0.790524778498846669317537792431, −0.31320372726410623386049221238,
0.31320372726410623386049221238, 0.790524778498846669317537792431, 1.78117217895741255629094341070, 2.54678111081278821672367778482, 2.60668542296021568850806672292, 3.26619671876105743918501083144, 3.83474277693139782937637541223, 4.49177206350173556172620669691, 4.90558428126005564635363790841, 5.24840843080219393777388302118, 5.48798976383435345079509547902, 6.08885116421139115631609832648, 6.51588379207177079750209058628, 7.23897220155182911351228848108, 7.41233094473798580137022766887, 7.87274201515483449585722434233, 8.392355685733742628052314985348, 8.431336248087487117452259236269, 8.876294685338878260001849519074, 9.529517495869652014777545323572
