| L(s) = 1 | + 4-s − 3·9-s − 6·11-s + 16·19-s − 12·29-s − 16·31-s − 3·36-s + 12·41-s − 6·44-s + 2·49-s − 18·59-s − 16·61-s − 64-s − 24·71-s + 16·76-s + 16·79-s + 36·89-s + 18·99-s − 24·101-s − 8·109-s − 12·116-s + 31·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s − 1.80·11-s + 3.67·19-s − 2.22·29-s − 2.87·31-s − 1/2·36-s + 1.87·41-s − 0.904·44-s + 2/7·49-s − 2.34·59-s − 2.04·61-s − 1/8·64-s − 2.84·71-s + 1.83·76-s + 1.80·79-s + 3.81·89-s + 1.80·99-s − 2.38·101-s − 0.766·109-s − 1.11·116-s + 2.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9409411717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9409411717\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 50 T^{2} + 291 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 145 T^{2} + 11616 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.937856997430255495321165883651, −7.74581612966021127894014292688, −7.48156421147113461603827302648, −7.27748430076572987679496070126, −7.24747657830854254685185883887, −7.18046604038872709637514970533, −6.37442517262936953245878493036, −6.19409333673517089030868964211, −5.93456960524839809523389131597, −5.77776783729059045135028451623, −5.43957702713819270948110051389, −5.28653110991761233435234100526, −5.19056635732825254653682358924, −4.87081096388530005545633985034, −4.31245964003600668225425439618, −4.20401803572603937524063952714, −3.48061156615724392144904788543, −3.36143391532898768042897988904, −3.25125020821996948147486049315, −2.92870463312217593216902191729, −2.41769554535967638132457069447, −2.24585239739276046473037093398, −1.60666082521912135561151358721, −1.32838530475066507362681787699, −0.33534494210286195735546655434,
0.33534494210286195735546655434, 1.32838530475066507362681787699, 1.60666082521912135561151358721, 2.24585239739276046473037093398, 2.41769554535967638132457069447, 2.92870463312217593216902191729, 3.25125020821996948147486049315, 3.36143391532898768042897988904, 3.48061156615724392144904788543, 4.20401803572603937524063952714, 4.31245964003600668225425439618, 4.87081096388530005545633985034, 5.19056635732825254653682358924, 5.28653110991761233435234100526, 5.43957702713819270948110051389, 5.77776783729059045135028451623, 5.93456960524839809523389131597, 6.19409333673517089030868964211, 6.37442517262936953245878493036, 7.18046604038872709637514970533, 7.24747657830854254685185883887, 7.27748430076572987679496070126, 7.48156421147113461603827302648, 7.74581612966021127894014292688, 7.937856997430255495321165883651
