| L(s) = 1 | + 2·5-s − 9-s − 16·19-s − 25-s + 12·29-s + 16·31-s + 12·41-s − 2·45-s − 2·49-s − 12·61-s + 32·71-s − 16·79-s + 81-s + 20·89-s − 32·95-s + 28·101-s + 20·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 3.67·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s + 1.87·41-s − 0.298·45-s − 2/7·49-s − 1.53·61-s + 3.79·71-s − 1.80·79-s + 1/9·81-s + 2.11·89-s − 3.28·95-s + 2.78·101-s + 1.91·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818502763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818502763\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.06399583628769522348420391225, −10.60034612058363192158274200886, −10.43933346297604854637444788363, −9.922414295394444478501342729305, −9.553699628350677336208885925654, −8.717041992659030265231901024847, −8.706711413662721585187756152851, −8.121187585531317711814726417526, −7.81086179628782486311194027225, −6.82159772057666119002449837992, −6.35466681614822261518805656337, −6.28535165740149600550140742612, −5.85715080165040228572494174744, −4.74413821504532104442114863720, −4.65534430363943296300570131916, −4.09900462851741552903729111613, −3.12495396796133784684915050699, −2.33571097080482315522199966525, −2.18094411208495541001292264158, −0.832206649013151396326080550604,
0.832206649013151396326080550604, 2.18094411208495541001292264158, 2.33571097080482315522199966525, 3.12495396796133784684915050699, 4.09900462851741552903729111613, 4.65534430363943296300570131916, 4.74413821504532104442114863720, 5.85715080165040228572494174744, 6.28535165740149600550140742612, 6.35466681614822261518805656337, 6.82159772057666119002449837992, 7.81086179628782486311194027225, 8.121187585531317711814726417526, 8.706711413662721585187756152851, 8.717041992659030265231901024847, 9.553699628350677336208885925654, 9.922414295394444478501342729305, 10.43933346297604854637444788363, 10.60034612058363192158274200886, 11.06399583628769522348420391225
