| L(s) = 1 | + 2·5-s + 10·17-s + 11·25-s − 32·29-s + 10·37-s − 16·41-s − 2·49-s − 2·53-s − 22·61-s − 30·73-s + 20·85-s + 14·89-s + 48·97-s − 22·101-s + 6·109-s − 16·113-s − 5·121-s + 38·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.42·17-s + 11/5·25-s − 5.94·29-s + 1.64·37-s − 2.49·41-s − 2/7·49-s − 0.274·53-s − 2.81·61-s − 3.51·73-s + 2.16·85-s + 1.48·89-s + 4.87·97-s − 2.18·101-s + 0.574·109-s − 1.50·113-s − 0.454·121-s + 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7209966255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7209966255\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 35 T^{2} + 864 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.53886626359238069758819959083, −5.98867641304559880921677687964, −5.98052437986808051965129298516, −5.97135009875434684985411415504, −5.89024236579650227554949090805, −5.39589148667742792058796890741, −5.19094015297682936207672878547, −5.11709915256627128873215644594, −4.93273931025385622770627041957, −4.72758042164038196313092923396, −4.25784684331809690121112509820, −4.17706867697275217776495393053, −3.61313289638765907186028980842, −3.60854335497047465632717484046, −3.58138971344786329739863231017, −3.08382444723340021941676418257, −2.95597875702242550220827768571, −2.80091824956010040611232995517, −2.27341304721973577030380357762, −1.95141787534657996090744182759, −1.72163364649437463951143090564, −1.57820317389206327820091561487, −1.27263606071932478773984107170, −0.852537544487195092694297287278, −0.13882210585428058306320824275,
0.13882210585428058306320824275, 0.852537544487195092694297287278, 1.27263606071932478773984107170, 1.57820317389206327820091561487, 1.72163364649437463951143090564, 1.95141787534657996090744182759, 2.27341304721973577030380357762, 2.80091824956010040611232995517, 2.95597875702242550220827768571, 3.08382444723340021941676418257, 3.58138971344786329739863231017, 3.60854335497047465632717484046, 3.61313289638765907186028980842, 4.17706867697275217776495393053, 4.25784684331809690121112509820, 4.72758042164038196313092923396, 4.93273931025385622770627041957, 5.11709915256627128873215644594, 5.19094015297682936207672878547, 5.39589148667742792058796890741, 5.89024236579650227554949090805, 5.97135009875434684985411415504, 5.98052437986808051965129298516, 5.98867641304559880921677687964, 6.53886626359238069758819959083
