| L(s) = 1 | − 4-s + 16-s − 16·19-s + 12·29-s − 8·31-s + 12·41-s − 49-s − 24·59-s − 20·61-s − 64-s − 24·71-s + 16·76-s − 16·79-s − 12·89-s − 12·101-s − 28·109-s − 12·116-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 3.67·19-s + 2.22·29-s − 1.43·31-s + 1.87·41-s − 1/7·49-s − 3.12·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 1.83·76-s − 1.80·79-s − 1.27·89-s − 1.19·101-s − 2.68·109-s − 1.11·116-s − 2·121-s + 0.718·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.394663982777576601807785052690, −8.339330707143783256402445841106, −7.79856665122842137546509773647, −7.46099889733317650240924556944, −6.85961694364785670576843052593, −6.68710253522566000263131383101, −6.08766621867549647746327852455, −5.90735893405611094874501195460, −5.67703750528507140179191063606, −4.64674453580542058632471879852, −4.53507047752588653882541464811, −4.43238676642480667306255207838, −3.95157743770693289972200245937, −3.22369915290844804845614088379, −2.77879340538732471404902232733, −2.43461805855537396249455206233, −1.65176570952002498107029789879, −1.34327336253784489312356493318, 0, 0,
1.34327336253784489312356493318, 1.65176570952002498107029789879, 2.43461805855537396249455206233, 2.77879340538732471404902232733, 3.22369915290844804845614088379, 3.95157743770693289972200245937, 4.43238676642480667306255207838, 4.53507047752588653882541464811, 4.64674453580542058632471879852, 5.67703750528507140179191063606, 5.90735893405611094874501195460, 6.08766621867549647746327852455, 6.68710253522566000263131383101, 6.85961694364785670576843052593, 7.46099889733317650240924556944, 7.79856665122842137546509773647, 8.339330707143783256402445841106, 8.394663982777576601807785052690
