| L(s) = 1 | − 2·5-s − 25-s + 20·29-s + 20·41-s + 14·49-s + 20·61-s − 20·89-s + 4·101-s − 12·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/5·25-s + 3.71·29-s + 3.12·41-s + 2·49-s + 2.56·61-s − 2.11·89-s + 0.398·101-s − 1.14·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049784379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049784379\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−9.654155887792734734389719725953, −9.420844271631511952858485504146, −8.724196582595219545465983625642, −8.534964065312342776141551775522, −8.162169145789519053179197596571, −7.79941900119667763682971898495, −7.22451300132872997519292084108, −7.07682236961043110027976443929, −6.34110085891932730965192004051, −6.26634789343380574013748111604, −5.45167070235756735737882026857, −5.27114339318429747720415150274, −4.40214920062796692947090991854, −4.27324654952734499041190090775, −3.93107504365283413400261443332, −3.11797907156419186966667701453, −2.65571018849977082810313264801, −2.30702506626782030106039441031, −1.10903140412871219835651823945, −0.70081637769018333116178111448,
0.70081637769018333116178111448, 1.10903140412871219835651823945, 2.30702506626782030106039441031, 2.65571018849977082810313264801, 3.11797907156419186966667701453, 3.93107504365283413400261443332, 4.27324654952734499041190090775, 4.40214920062796692947090991854, 5.27114339318429747720415150274, 5.45167070235756735737882026857, 6.26634789343380574013748111604, 6.34110085891932730965192004051, 7.07682236961043110027976443929, 7.22451300132872997519292084108, 7.79941900119667763682971898495, 8.162169145789519053179197596571, 8.534964065312342776141551775522, 8.724196582595219545465983625642, 9.420844271631511952858485504146, 9.654155887792734734389719725953
