| L(s) = 1 | − 2·5-s − 4·7-s − 3·9-s + 12·17-s + 3·25-s + 8·35-s − 4·37-s + 12·41-s + 16·43-s + 6·45-s + 24·47-s + 9·49-s + 24·59-s + 12·63-s − 16·67-s − 16·79-s + 9·81-s − 24·85-s + 12·89-s − 12·101-s − 4·109-s − 48·119-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 9-s + 2.91·17-s + 3/5·25-s + 1.35·35-s − 0.657·37-s + 1.87·41-s + 2.43·43-s + 0.894·45-s + 3.50·47-s + 9/7·49-s + 3.12·59-s + 1.51·63-s − 1.95·67-s − 1.80·79-s + 81-s − 2.60·85-s + 1.27·89-s − 1.19·101-s − 0.383·109-s − 4.40·119-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.701900254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.701900254\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.480696064539601407731812953031, −9.163571076100335377451718206971, −8.825883601165449380528531189619, −8.368801190515624475181747662042, −7.902039734334558043555125042796, −7.51555577643464559491547712804, −7.21463024851942921108605313917, −6.99531770856166885083132845989, −6.08534575734793081410296563567, −5.90498972589107391443652397480, −5.54839783082297697438556053599, −5.36558660343883412833402128804, −4.20038993135112679129094501772, −4.17110607161746032592721463974, −3.59652540692593678369843025719, −2.97299256042288683656230736871, −2.93633057642367625802991181019, −2.23283319459661454064838374185, −0.858340131635840093266484899711, −0.72206754198545490062980935577,
0.72206754198545490062980935577, 0.858340131635840093266484899711, 2.23283319459661454064838374185, 2.93633057642367625802991181019, 2.97299256042288683656230736871, 3.59652540692593678369843025719, 4.17110607161746032592721463974, 4.20038993135112679129094501772, 5.36558660343883412833402128804, 5.54839783082297697438556053599, 5.90498972589107391443652397480, 6.08534575734793081410296563567, 6.99531770856166885083132845989, 7.21463024851942921108605313917, 7.51555577643464559491547712804, 7.902039734334558043555125042796, 8.368801190515624475181747662042, 8.825883601165449380528531189619, 9.163571076100335377451718206971, 9.480696064539601407731812953031
