| L(s) = 1 | + 16·3-s + 128·9-s − 128·16-s + 216·23-s − 74·25-s + 592·27-s + 868·37-s − 72·47-s − 2.04e3·48-s + 1.47e3·53-s − 832·67-s + 3.45e3·69-s + 2.44e3·71-s − 1.18e3·75-s − 78·81-s + 68·97-s − 2.34e3·103-s + 1.38e4·111-s − 4.28e3·113-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.15e3·141-s − 1.63e4·144-s + 149-s + ⋯ |
| L(s) = 1 | + 3.07·3-s + 4.74·9-s − 2·16-s + 1.95·23-s − 0.591·25-s + 4.21·27-s + 3.85·37-s − 0.223·47-s − 6.15·48-s + 3.82·53-s − 1.51·67-s + 6.02·69-s + 4.09·71-s − 1.82·75-s − 0.106·81-s + 0.0711·97-s − 2.24·103-s + 11.8·111-s − 3.56·113-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.688·141-s − 9.48·144-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.109706118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.109706118\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 + 74 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )^{2}( 1 + p^{2} T + p^{3} T^{2} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2}( 1 + 10 T^{2} + p^{6} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2}( 1 - 12670 T^{2} + p^{6} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 56018 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{2}( 1 + 87050 T^{2} + p^{6} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2}( 1 - 206350 T^{2} + p^{6} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - 738 T + p^{3} T^{2} )^{2}( 1 + 246890 T^{2} + p^{6} T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2}( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 416 T + p^{3} T^{2} )^{2}( 1 - 428470 T^{2} + p^{6} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 612 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 1392338 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2}( 1 - 1824190 T^{2} + p^{6} T^{4} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80202025866611705450800240344, −10.69701681766516781198101206465, −9.950840444882754937550589140967, −9.704207867856288971683154936283, −9.550721084115330526260482703222, −9.198586970421434548627355019533, −8.888210704110760354256284006288, −8.866827352528231100260621137280, −8.344144672474436215240966711005, −8.092629321399285943733370139222, −7.72075572202395053313864480591, −7.55154191761109222397095255468, −7.00017714089702681497529406841, −6.72123278264878267161151112493, −6.50076156725777977573451644176, −5.49786431894925936180304482010, −5.48921593299088296705941566786, −4.45200529896011079006936054124, −4.33689676640315213165640614361, −3.95462466167787870723331052035, −3.30037732998850853676709861136, −2.80875081885556970428058747606, −2.35467146084737596663313835140, −2.27968823903062374872360117555, −1.02868986916054655025195082081,
1.02868986916054655025195082081, 2.27968823903062374872360117555, 2.35467146084737596663313835140, 2.80875081885556970428058747606, 3.30037732998850853676709861136, 3.95462466167787870723331052035, 4.33689676640315213165640614361, 4.45200529896011079006936054124, 5.48921593299088296705941566786, 5.49786431894925936180304482010, 6.50076156725777977573451644176, 6.72123278264878267161151112493, 7.00017714089702681497529406841, 7.55154191761109222397095255468, 7.72075572202395053313864480591, 8.092629321399285943733370139222, 8.344144672474436215240966711005, 8.866827352528231100260621137280, 8.888210704110760354256284006288, 9.198586970421434548627355019533, 9.550721084115330526260482703222, 9.704207867856288971683154936283, 9.950840444882754937550589140967, 10.69701681766516781198101206465, 10.80202025866611705450800240344
