| L(s) = 1 | + 6·5-s − 5·9-s + 17·25-s + 14·37-s − 30·45-s − 14·49-s + 12·53-s + 16·81-s − 18·89-s − 34·97-s − 42·113-s − 11·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 84·185-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 5/3·9-s + 17/5·25-s + 2.30·37-s − 4.47·45-s − 2·49-s + 1.64·53-s + 16/9·81-s − 1.90·89-s − 3.45·97-s − 3.95·113-s − 121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 6.17·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.738457921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738457921\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−13.26689574460251639152912224510, −12.66488246036728006839149801352, −11.85656608214118645293502039957, −11.49198379525622179553040582408, −10.81544137256835978847799455611, −10.51289893843233578366461806575, −9.748489922897669210009610733974, −9.412776763383933013265662140945, −9.295598080071927005493129245652, −8.265812388964835060097532459611, −8.200933396978128641688989042432, −7.02685845751682152482628040032, −6.36473958059617845155195121868, −6.04770729300202012507919444525, −5.46736213803717728599297204420, −5.26289601900205623389920953260, −4.14541389254925606064779588488, −2.79614747922257642985836490734, −2.57926410663713843923971601300, −1.54947310926681241241979002733,
1.54947310926681241241979002733, 2.57926410663713843923971601300, 2.79614747922257642985836490734, 4.14541389254925606064779588488, 5.26289601900205623389920953260, 5.46736213803717728599297204420, 6.04770729300202012507919444525, 6.36473958059617845155195121868, 7.02685845751682152482628040032, 8.200933396978128641688989042432, 8.265812388964835060097532459611, 9.295598080071927005493129245652, 9.412776763383933013265662140945, 9.748489922897669210009610733974, 10.51289893843233578366461806575, 10.81544137256835978847799455611, 11.49198379525622179553040582408, 11.85656608214118645293502039957, 12.66488246036728006839149801352, 13.26689574460251639152912224510
