| L(s) = 1 | − 3·3-s + 5-s + 2·7-s + 4·9-s + 11-s + 13-s − 3·15-s − 12·17-s − 4·19-s − 6·21-s − 3·23-s − 6·25-s − 6·27-s − 3·29-s + 3·31-s − 3·33-s + 2·35-s − 2·37-s − 3·39-s + 9·41-s + 6·43-s + 4·45-s + 2·47-s + 2·49-s + 36·51-s + 12·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.755·7-s + 4/3·9-s + 0.301·11-s + 0.277·13-s − 0.774·15-s − 2.91·17-s − 0.917·19-s − 1.30·21-s − 0.625·23-s − 6/5·25-s − 1.15·27-s − 0.557·29-s + 0.538·31-s − 0.522·33-s + 0.338·35-s − 0.328·37-s − 0.480·39-s + 1.40·41-s + 0.914·43-s + 0.596·45-s + 0.291·47-s + 2/7·49-s + 5.04·51-s + 1.64·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 83 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 253 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.985017198244569993205861149383, −8.488195384857996670944557521108, −7.985610171977074464649569877318, −7.39662100747186063632623998386, −7.32088028196587282419508517849, −6.55753432047408324251295934864, −6.37134261285009509756808409962, −6.10223774884279333131430876112, −5.63076875998154267247011087921, −5.46046707529506985651634671878, −4.76280545299010122828466146824, −4.41933973972685060954460496120, −4.03968676816200616349597357438, −3.92677720540305487400794133529, −2.52088968681583352688459548288, −2.48542401981367028793302196799, −1.73378050295571585239151369541, −1.28093493432064392838855926278, 0, 0,
1.28093493432064392838855926278, 1.73378050295571585239151369541, 2.48542401981367028793302196799, 2.52088968681583352688459548288, 3.92677720540305487400794133529, 4.03968676816200616349597357438, 4.41933973972685060954460496120, 4.76280545299010122828466146824, 5.46046707529506985651634671878, 5.63076875998154267247011087921, 6.10223774884279333131430876112, 6.37134261285009509756808409962, 6.55753432047408324251295934864, 7.32088028196587282419508517849, 7.39662100747186063632623998386, 7.985610171977074464649569877318, 8.488195384857996670944557521108, 8.985017198244569993205861149383
