| L(s) = 1 | − 4-s − 5-s + 7-s − 11-s − 13-s − 2·17-s + 20-s − 28-s + 2·29-s − 35-s + 44-s + 47-s + 52-s + 55-s + 64-s + 65-s + 2·68-s + 2·71-s + 2·73-s − 77-s + 79-s + 83-s + 2·85-s − 91-s − 97-s + 2·103-s − 2·109-s + ⋯ |
| L(s) = 1 | − 4-s − 5-s + 7-s − 11-s − 13-s − 2·17-s + 20-s − 28-s + 2·29-s − 35-s + 44-s + 47-s + 52-s + 55-s + 64-s + 65-s + 2·68-s + 2·71-s + 2·73-s − 77-s + 79-s + 83-s + 2·85-s − 91-s − 97-s + 2·103-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4636521176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4636521176\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−10.82917599296387125745317909102, −10.02974132397202169235033174427, −9.532782310488458177315716868608, −9.186666978843783051389022427274, −8.695906947112781539843189901559, −8.214437140811042854537579808206, −8.192409724254205467234083184664, −7.62295267227576626149145311636, −7.23615400486247066909131052547, −6.52102065956943462282851621884, −6.46014357494292055649862723112, −5.27029414542421131394738961681, −5.17572676929868189460097534854, −4.70674051537928115428115562533, −4.30860543349632076640617416905, −4.00835789159501541687384154746, −3.19163578160091161782215306127, −2.38865482046708246309519136320, −2.13296047652412830308913082438, −0.66210645844567043654430093127,
0.66210645844567043654430093127, 2.13296047652412830308913082438, 2.38865482046708246309519136320, 3.19163578160091161782215306127, 4.00835789159501541687384154746, 4.30860543349632076640617416905, 4.70674051537928115428115562533, 5.17572676929868189460097534854, 5.27029414542421131394738961681, 6.46014357494292055649862723112, 6.52102065956943462282851621884, 7.23615400486247066909131052547, 7.62295267227576626149145311636, 8.192409724254205467234083184664, 8.214437140811042854537579808206, 8.695906947112781539843189901559, 9.186666978843783051389022427274, 9.532782310488458177315716868608, 10.02974132397202169235033174427, 10.82917599296387125745317909102
