| L(s) = 1 | − 9-s + 12·11-s − 10·19-s − 12·29-s + 2·31-s + 13·49-s + 12·59-s + 26·61-s + 16·79-s + 81-s − 12·99-s + 24·101-s − 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 10·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.61·11-s − 2.29·19-s − 2.22·29-s + 0.359·31-s + 13/7·49-s + 1.56·59-s + 3.32·61-s + 1.80·79-s + 1/9·81-s − 1.20·99-s + 2.38·101-s − 1.34·109-s + 7.81·121-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 + 0.0773·167-s + 1/13·169-s + 0.764·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.680709707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.680709707\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.452293957893836008865088767680, −8.327226487992534623057315491699, −7.79269256947370971533460886624, −7.05039447424064324140924532244, −6.92172840405867299956095063243, −6.85818785654417770486200675013, −6.22039407250702694528961799349, −6.07951121888303215833571830546, −5.65478699821192405326711161179, −5.25983896758101428045009510703, −4.50940263472266057808710524758, −4.30564465930205384807314079011, −3.82997973735432705378852402607, −3.75105380915317385580510289194, −3.39873202620584695147564440633, −2.42766151147122454065430182708, −2.01728678140894120673065218438, −1.84873406468707104252775814054, −0.998007505833307047385853231120, −0.61784650159165587218007995658,
0.61784650159165587218007995658, 0.998007505833307047385853231120, 1.84873406468707104252775814054, 2.01728678140894120673065218438, 2.42766151147122454065430182708, 3.39873202620584695147564440633, 3.75105380915317385580510289194, 3.82997973735432705378852402607, 4.30564465930205384807314079011, 4.50940263472266057808710524758, 5.25983896758101428045009510703, 5.65478699821192405326711161179, 6.07951121888303215833571830546, 6.22039407250702694528961799349, 6.85818785654417770486200675013, 6.92172840405867299956095063243, 7.05039447424064324140924532244, 7.79269256947370971533460886624, 8.327226487992534623057315491699, 8.452293957893836008865088767680
