| L(s) = 1 | + 3-s − 3·5-s − 2·7-s − 9-s − 2·13-s − 3·15-s − 4·17-s + 8·19-s − 2·21-s − 9·23-s + 25-s − 2·29-s + 7·31-s + 6·35-s + 11·37-s − 2·39-s − 6·41-s + 6·43-s + 3·45-s − 16·47-s + 6·49-s − 4·51-s − 8·53-s + 8·57-s − 5·59-s − 6·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.755·7-s − 1/3·9-s − 0.554·13-s − 0.774·15-s − 0.970·17-s + 1.83·19-s − 0.436·21-s − 1.87·23-s + 1/5·25-s − 0.371·29-s + 1.25·31-s + 1.01·35-s + 1.80·37-s − 0.320·39-s − 0.937·41-s + 0.914·43-s + 0.447·45-s − 2.33·47-s + 6/7·49-s − 0.560·51-s − 1.09·53-s + 1.05·57-s − 0.650·59-s − 0.768·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 27 T + 372 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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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
−7.71012333098053052243334732074, −7.67200345678067202383208474597, −6.95391682314268620118293528732, −6.79036160709965208373862507876, −6.26968558320890098744015448234, −6.09392548294987421341637371512, −5.63728208451714129528673518946, −5.17936817038298565910034740892, −4.75013249513086591301762770704, −4.31589894052272916640617190924, −4.19624347706959284344875058661, −3.63640730214586215705516311289, −3.18840850231594885172768986287, −3.15116864201567348428547078141, −2.49210957468196495633600849714, −2.24803446888915899261214553519, −1.51611110382579419061686343950, −0.914593848108578311340633650519, 0, 0,
0.914593848108578311340633650519, 1.51611110382579419061686343950, 2.24803446888915899261214553519, 2.49210957468196495633600849714, 3.15116864201567348428547078141, 3.18840850231594885172768986287, 3.63640730214586215705516311289, 4.19624347706959284344875058661, 4.31589894052272916640617190924, 4.75013249513086591301762770704, 5.17936817038298565910034740892, 5.63728208451714129528673518946, 6.09392548294987421341637371512, 6.26968558320890098744015448234, 6.79036160709965208373862507876, 6.95391682314268620118293528732, 7.67200345678067202383208474597, 7.71012333098053052243334732074
