| L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·15-s − 25-s − 27-s − 2·45-s + 10·49-s − 20·53-s + 75-s + 16·79-s + 81-s + 8·83-s + 8·107-s + 18·121-s + 12·125-s + 127-s + 131-s + 2·135-s + 137-s + 139-s − 10·147-s + 149-s + 151-s + 157-s + 20·159-s + 163-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s − 1/5·25-s − 0.192·27-s − 0.298·45-s + 10/7·49-s − 2.74·53-s + 0.115·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s + 0.773·107-s + 1.63·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s − 0.824·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.58·159-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9176064180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9176064180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.720556122204919940787114226817, −8.127280331995193708455482208382, −7.85101984287289917616469398437, −7.34999160539775999043577503268, −6.95203567399306177371933355135, −6.31795351810838792684187487407, −5.99939825891420261041909475864, −5.37595707552247311413898151676, −4.76079991274913578801014190589, −4.46360011835707575831995568528, −3.70428071362386722458673170717, −3.37528171703364868655624815248, −2.48435940156910725970454962683, −1.65720388043715485575795658732, −0.57100453902310593927136460717,
0.57100453902310593927136460717, 1.65720388043715485575795658732, 2.48435940156910725970454962683, 3.37528171703364868655624815248, 3.70428071362386722458673170717, 4.46360011835707575831995568528, 4.76079991274913578801014190589, 5.37595707552247311413898151676, 5.99939825891420261041909475864, 6.31795351810838792684187487407, 6.95203567399306177371933355135, 7.34999160539775999043577503268, 7.85101984287289917616469398437, 8.127280331995193708455482208382, 8.720556122204919940787114226817
