| L(s) = 1 | − 2·7-s + 8·13-s + 12·19-s − 4·25-s − 6·31-s + 8·37-s − 4·43-s − 10·49-s + 16·61-s + 8·67-s − 2·79-s − 16·91-s + 8·97-s + 14·103-s + 16·109-s + 2·121-s + 127-s + 131-s − 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2.21·13-s + 2.75·19-s − 4/5·25-s − 1.07·31-s + 1.31·37-s − 0.609·43-s − 1.42·49-s + 2.04·61-s + 0.977·67-s − 0.225·79-s − 1.67·91-s + 0.812·97-s + 1.37·103-s + 1.53·109-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-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 + 0.0773·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384600830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384600830\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.035411896936685983598566680489, −7.43328684510365823872466754091, −7.24848644122997854173655428926, −6.53457211549882167634017441133, −6.22894001515526176073695411055, −5.86147448338780147680621672471, −5.34106585077350778108562006021, −5.04040595207224797404037466404, −4.20071953031454820929806675687, −3.65055031433153528531885833664, −3.42268573477489510493362822498, −3.01837725983234097025204450396, −2.10587414943305469446637052298, −1.35056656441704338145793774049, −0.75681043524687564940132265064,
0.75681043524687564940132265064, 1.35056656441704338145793774049, 2.10587414943305469446637052298, 3.01837725983234097025204450396, 3.42268573477489510493362822498, 3.65055031433153528531885833664, 4.20071953031454820929806675687, 5.04040595207224797404037466404, 5.34106585077350778108562006021, 5.86147448338780147680621672471, 6.22894001515526176073695411055, 6.53457211549882167634017441133, 7.24848644122997854173655428926, 7.43328684510365823872466754091, 8.035411896936685983598566680489
