| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s + 2·12-s − 12·13-s + 16-s + 3·18-s + 2·24-s − 12·26-s + 4·27-s − 16·31-s + 32-s + 3·36-s − 4·37-s − 24·39-s + 4·41-s + 8·43-s + 2·48-s − 10·49-s − 12·52-s − 12·53-s + 4·54-s − 16·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s + 0.577·12-s − 3.32·13-s + 1/4·16-s + 0.707·18-s + 0.408·24-s − 2.35·26-s + 0.769·27-s − 2.87·31-s + 0.176·32-s + 1/2·36-s − 0.657·37-s − 3.84·39-s + 0.624·41-s + 1.21·43-s + 0.288·48-s − 1.42·49-s − 1.66·52-s − 1.64·53-s + 0.544·54-s − 2.03·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.82425125324178640900965713696, −7.50083844850222126920204719803, −7.47947512213883760565427163251, −6.64959962043719359186062300669, −6.62229130807727725313447057117, −5.51306797780401171163763542733, −5.13544146251726925083565635926, −4.98557453494780998363680778315, −4.21329681476984441450446373250, −3.79143046769365773246334925917, −3.22247217757406603059726753652, −2.45815773509881651106722995929, −2.36879858490653300083733004583, −1.62764325671716933168154276530, 0,
1.62764325671716933168154276530, 2.36879858490653300083733004583, 2.45815773509881651106722995929, 3.22247217757406603059726753652, 3.79143046769365773246334925917, 4.21329681476984441450446373250, 4.98557453494780998363680778315, 5.13544146251726925083565635926, 5.51306797780401171163763542733, 6.62229130807727725313447057117, 6.64959962043719359186062300669, 7.47947512213883760565427163251, 7.50083844850222126920204719803, 7.82425125324178640900965713696
