| L(s) = 1 | − 4-s − 9-s + 12·11-s + 16-s − 8·19-s − 4·31-s + 36-s + 10·41-s − 12·44-s + 14·49-s + 6·59-s + 10·61-s − 64-s + 10·71-s + 8·76-s + 16·79-s + 81-s − 28·89-s − 12·99-s + 20·101-s + 20·109-s + 86·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 3.61·11-s + 1/4·16-s − 1.83·19-s − 0.718·31-s + 1/6·36-s + 1.56·41-s − 1.80·44-s + 2·49-s + 0.781·59-s + 1.28·61-s − 1/8·64-s + 1.18·71-s + 0.917·76-s + 1.80·79-s + 1/9·81-s − 2.96·89-s − 1.20·99-s + 1.99·101-s + 1.91·109-s + 7.81·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.980136427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.980136427\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.029613404362900978178823490171, −8.861227858636236296327715831911, −8.404287303391963351279336309429, −8.210294379297691321538279326149, −7.37579653310079361538377602996, −7.21132089484149374748748345031, −6.65418256735679247962642165776, −6.35024803787895492736169678760, −6.24335644334790230713211404676, −5.56892643260902236513624021552, −5.31373220415078402401210557398, −4.42828915703440611846751360443, −4.19798728007053181961321240864, −4.02576201808408322378946990158, −3.62986185744889399820409105794, −3.05863375921527275644362115194, −2.10301667276574041648777851317, −1.99834101287055966919933451227, −1.07733731989298990584220617126, −0.69845961306140411679436682715,
0.69845961306140411679436682715, 1.07733731989298990584220617126, 1.99834101287055966919933451227, 2.10301667276574041648777851317, 3.05863375921527275644362115194, 3.62986185744889399820409105794, 4.02576201808408322378946990158, 4.19798728007053181961321240864, 4.42828915703440611846751360443, 5.31373220415078402401210557398, 5.56892643260902236513624021552, 6.24335644334790230713211404676, 6.35024803787895492736169678760, 6.65418256735679247962642165776, 7.21132089484149374748748345031, 7.37579653310079361538377602996, 8.210294379297691321538279326149, 8.404287303391963351279336309429, 8.861227858636236296327715831911, 9.029613404362900978178823490171
