| L(s) = 1 | + 2·4-s − 2·7-s + 13-s + 3·16-s + 19-s − 25-s − 4·28-s + 31-s − 2·37-s + 2·43-s + 49-s + 2·52-s + 61-s + 4·64-s − 2·67-s − 2·73-s + 2·76-s + 79-s − 2·91-s − 2·97-s − 2·100-s + 103-s − 2·109-s − 6·112-s + 2·121-s + 2·124-s + 127-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·7-s + 13-s + 3·16-s + 19-s − 25-s − 4·28-s + 31-s − 2·37-s + 2·43-s + 49-s + 2·52-s + 61-s + 4·64-s − 2·67-s − 2·73-s + 2·76-s + 79-s − 2·91-s − 2·97-s − 2·100-s + 103-s − 2·109-s − 6·112-s + 2·121-s + 2·124-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1347921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1347921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.463765163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463765163\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.11779515064140297183398330842, −10.07419572051567504980365840619, −9.280473487018972128925589942959, −9.244806149781125474964387460467, −8.482392297259216021734387067696, −8.031569500144506926581163831438, −7.59373121520222109130086504021, −7.19246982468968582739430830558, −6.70031291078697060860902001831, −6.63446195978343204296019784070, −6.01420069312561057990132415775, −5.75157735074639535605728152664, −5.49943245078098159310286938224, −4.49759048021314062758115443788, −3.67780101629692575630618606475, −3.53682757381979730556938540778, −2.90426503836505429908136918426, −2.69401757824351624191926031953, −1.83565347394534063440904774887, −1.17448564662090297132054259887,
1.17448564662090297132054259887, 1.83565347394534063440904774887, 2.69401757824351624191926031953, 2.90426503836505429908136918426, 3.53682757381979730556938540778, 3.67780101629692575630618606475, 4.49759048021314062758115443788, 5.49943245078098159310286938224, 5.75157735074639535605728152664, 6.01420069312561057990132415775, 6.63446195978343204296019784070, 6.70031291078697060860902001831, 7.19246982468968582739430830558, 7.59373121520222109130086504021, 8.031569500144506926581163831438, 8.482392297259216021734387067696, 9.244806149781125474964387460467, 9.280473487018972128925589942959, 10.07419572051567504980365840619, 10.11779515064140297183398330842
