| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 2·7-s + 4·10-s + 6·11-s − 4·14-s − 4·16-s + 10·17-s − 10·19-s + 4·20-s + 12·22-s + 2·23-s − 25-s − 4·28-s − 10·29-s − 8·32-s + 20·34-s − 4·35-s − 20·38-s + 12·44-s + 4·46-s + 10·47-s + 2·49-s − 2·50-s + 12·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 0.755·7-s + 1.26·10-s + 1.80·11-s − 1.06·14-s − 16-s + 2.42·17-s − 2.29·19-s + 0.894·20-s + 2.55·22-s + 0.417·23-s − 1/5·25-s − 0.755·28-s − 1.85·29-s − 1.41·32-s + 3.42·34-s − 0.676·35-s − 3.24·38-s + 1.80·44-s + 0.589·46-s + 1.45·47-s + 2/7·49-s − 0.282·50-s + 1.64·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.853170928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.853170928\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.78818038008263163006320356212, −10.06844829734233093165981790435, −9.846486833442711328541478967324, −9.401156830394144132696027899325, −8.852194238098621979886645261755, −8.807915074025400598698948206934, −7.927731919808076296094650955924, −7.33255170325102814654137065386, −6.95238805493118761581891668602, −6.29508844161376618080859404297, −6.18443217349376686242264510652, −5.76033436385203747189479876603, −5.31304407486482475797447051310, −4.73136569614017688313532345477, −3.93441870436205427769687110698, −3.67746589975400401842350820282, −3.46472446487291869445330415489, −2.31437001524771722857797466328, −2.08588986796309718270366559953, −0.955975356322181337656235837107,
0.955975356322181337656235837107, 2.08588986796309718270366559953, 2.31437001524771722857797466328, 3.46472446487291869445330415489, 3.67746589975400401842350820282, 3.93441870436205427769687110698, 4.73136569614017688313532345477, 5.31304407486482475797447051310, 5.76033436385203747189479876603, 6.18443217349376686242264510652, 6.29508844161376618080859404297, 6.95238805493118761581891668602, 7.33255170325102814654137065386, 7.927731919808076296094650955924, 8.807915074025400598698948206934, 8.852194238098621979886645261755, 9.401156830394144132696027899325, 9.846486833442711328541478967324, 10.06844829734233093165981790435, 10.78818038008263163006320356212
