| L(s) = 1 | − 2·5-s + 2·7-s + 6·11-s − 7·25-s + 12·29-s + 14·31-s − 4·35-s − 11·49-s + 18·53-s − 12·55-s − 8·59-s − 6·73-s + 12·77-s − 8·79-s + 14·83-s + 14·97-s + 34·101-s + 6·107-s + 5·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.80·11-s − 7/5·25-s + 2.22·29-s + 2.51·31-s − 0.676·35-s − 1.57·49-s + 2.47·53-s − 1.61·55-s − 1.04·59-s − 0.702·73-s + 1.36·77-s − 0.900·79-s + 1.53·83-s + 1.42·97-s + 3.38·101-s + 0.580·107-s + 5/11·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.644329017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.644329017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−8.020381197661248698718192528167, −7.992542229919320830017389804362, −7.30432295425651012562502237548, −7.29703569452094638275288971023, −6.56178477552353195606235272832, −6.49395973570224721048713681020, −6.08715668148829898815562507988, −5.88268255963803482179124709230, −5.12098813437433552223115508905, −4.79993427475740711515106419482, −4.46258364134300092209527170135, −4.31403324474316440739449150089, −3.77843069004069771137691737819, −3.49385546113585354226210487085, −3.01898593901985181909277463273, −2.51605575010108787661477615008, −2.00484623905548808009679273348, −1.47536598950536411222855825324, −0.976856393918304537559425562636, −0.57285960439073657517597686290,
0.57285960439073657517597686290, 0.976856393918304537559425562636, 1.47536598950536411222855825324, 2.00484623905548808009679273348, 2.51605575010108787661477615008, 3.01898593901985181909277463273, 3.49385546113585354226210487085, 3.77843069004069771137691737819, 4.31403324474316440739449150089, 4.46258364134300092209527170135, 4.79993427475740711515106419482, 5.12098813437433552223115508905, 5.88268255963803482179124709230, 6.08715668148829898815562507988, 6.49395973570224721048713681020, 6.56178477552353195606235272832, 7.29703569452094638275288971023, 7.30432295425651012562502237548, 7.992542229919320830017389804362, 8.020381197661248698718192528167
