| L(s) = 1 | + 4·5-s + 4·11-s + 16·19-s + 11·25-s − 16·29-s − 4·41-s + 10·49-s + 16·55-s + 12·59-s − 4·61-s + 8·71-s + 16·79-s + 12·89-s + 64·95-s − 12·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.20·11-s + 3.67·19-s + 11/5·25-s − 2.97·29-s − 0.624·41-s + 10/7·49-s + 2.15·55-s + 1.56·59-s − 0.512·61-s + 0.949·71-s + 1.80·79-s + 1.27·89-s + 6.56·95-s − 1.14·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.582345327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.582345327\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.158011750354362006742613272191, −8.980020917645437582523382568979, −8.082861417968255847682690023734, −7.80054338687082878417070857827, −7.43578674160238268802876432587, −6.94788724628019218087708838317, −6.76145677851338097963760276451, −6.30820571926707634797879843440, −5.66002179569545586628642043882, −5.54506790601799216868597445911, −5.29069678724686815112295690940, −4.98839706083225472542326402310, −4.04366971032809460292697041116, −3.85402845234041621627032617068, −3.15199237533902869237158795073, −3.05015168026851837114264694731, −2.02339348374356193843564255193, −1.98278026489574169808214591022, −1.18449693014667912075554981992, −0.868132391018211758700877080784,
0.868132391018211758700877080784, 1.18449693014667912075554981992, 1.98278026489574169808214591022, 2.02339348374356193843564255193, 3.05015168026851837114264694731, 3.15199237533902869237158795073, 3.85402845234041621627032617068, 4.04366971032809460292697041116, 4.98839706083225472542326402310, 5.29069678724686815112295690940, 5.54506790601799216868597445911, 5.66002179569545586628642043882, 6.30820571926707634797879843440, 6.76145677851338097963760276451, 6.94788724628019218087708838317, 7.43578674160238268802876432587, 7.80054338687082878417070857827, 8.082861417968255847682690023734, 8.980020917645437582523382568979, 9.158011750354362006742613272191
