| L(s) = 1 | + 2·3-s + 3·9-s − 12·17-s − 8·19-s − 10·25-s + 4·27-s + 20·31-s + 2·49-s − 24·51-s − 16·57-s + 24·59-s + 20·61-s + 8·67-s + 24·71-s − 4·73-s − 20·75-s − 20·79-s + 5·81-s + 40·93-s + 4·103-s + 24·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.91·17-s − 1.83·19-s − 2·25-s + 0.769·27-s + 3.59·31-s + 2/7·49-s − 3.36·51-s − 2.11·57-s + 3.12·59-s + 2.56·61-s + 0.977·67-s + 2.84·71-s − 0.468·73-s − 2.30·75-s − 2.25·79-s + 5/9·81-s + 4.14·93-s + 0.394·103-s + 2.32·107-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.329·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.458312075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458312075\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−10.08334375561979251765025757226, −9.928893733422083180700432704864, −9.533102319122955538459747419432, −8.763484474856935053271925669620, −8.599715891584782800875989555332, −8.356909299591533214093019428851, −8.068998870698734215836816043440, −7.33865216613293070777849546123, −6.74179197046304684145964181699, −6.61005108446393598455291668793, −6.25390693417495342199817470878, −5.50037755594958602220893982706, −4.79120571923616587209650478490, −4.21364101527233722140472561117, −4.20194994975886507588875225753, −3.58786659845688741694869860454, −2.54561658148014300309382015204, −2.30900102770700837006684079715, −2.04358279692190287755073988092, −0.66959445992424545105962945263,
0.66959445992424545105962945263, 2.04358279692190287755073988092, 2.30900102770700837006684079715, 2.54561658148014300309382015204, 3.58786659845688741694869860454, 4.20194994975886507588875225753, 4.21364101527233722140472561117, 4.79120571923616587209650478490, 5.50037755594958602220893982706, 6.25390693417495342199817470878, 6.61005108446393598455291668793, 6.74179197046304684145964181699, 7.33865216613293070777849546123, 8.068998870698734215836816043440, 8.356909299591533214093019428851, 8.599715891584782800875989555332, 8.763484474856935053271925669620, 9.533102319122955538459747419432, 9.928893733422083180700432704864, 10.08334375561979251765025757226
