| L(s) = 1 | − 5-s + 5·7-s + 6·11-s + 4·13-s − 6·17-s − 8·19-s + 3·23-s − 6·29-s − 2·31-s − 5·35-s − 8·37-s + 6·41-s + 10·43-s + 18·49-s + 12·53-s − 6·55-s + 61-s − 4·65-s + 7·67-s + 10·73-s + 30·77-s + 4·79-s − 6·83-s + 6·85-s − 3·89-s + 20·91-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.88·7-s + 1.80·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 0.625·23-s − 1.11·29-s − 0.359·31-s − 0.845·35-s − 1.31·37-s + 0.937·41-s + 1.52·43-s + 18/7·49-s + 1.64·53-s − 0.809·55-s + 0.128·61-s − 0.496·65-s + 0.855·67-s + 1.17·73-s + 3.41·77-s + 0.450·79-s − 0.658·83-s + 0.650·85-s − 0.317·89-s + 2.09·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.888152887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888152887\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.830876611086820068999627703096, −9.126428047352636703313038229226, −9.024932152330816997235176021680, −8.754252631221316689724545015977, −8.245006239745881421443089921481, −8.120293836222662568042044607843, −7.34278921995405715382349004648, −7.02161594270452906245617475796, −6.69817117633524420057865160748, −6.18881934609651796150794330490, −5.67457652762456725804046782168, −5.29080510672150683254101321524, −4.54780852555171622326869929089, −4.21278949124244672537288753854, −4.01258262723753373635712756109, −3.59137872647668111989486810239, −2.48677459122370446479614714613, −1.98989712732978514722706053451, −1.54599485178362962683927342140, −0.76542102191488171360569893374,
0.76542102191488171360569893374, 1.54599485178362962683927342140, 1.98989712732978514722706053451, 2.48677459122370446479614714613, 3.59137872647668111989486810239, 4.01258262723753373635712756109, 4.21278949124244672537288753854, 4.54780852555171622326869929089, 5.29080510672150683254101321524, 5.67457652762456725804046782168, 6.18881934609651796150794330490, 6.69817117633524420057865160748, 7.02161594270452906245617475796, 7.34278921995405715382349004648, 8.120293836222662568042044607843, 8.245006239745881421443089921481, 8.754252631221316689724545015977, 9.024932152330816997235176021680, 9.126428047352636703313038229226, 9.830876611086820068999627703096
