| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 4·13-s + 16-s − 2·17-s − 18-s − 10·25-s − 4·26-s − 32-s + 2·34-s + 36-s + 16·37-s + 12·41-s − 10·49-s + 10·50-s + 4·52-s + 12·53-s + 16·61-s + 64-s − 2·68-s − 72-s + 4·73-s − 16·74-s + 81-s − 12·82-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 2·25-s − 0.784·26-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 2.63·37-s + 1.87·41-s − 1.42·49-s + 1.41·50-s + 0.554·52-s + 1.64·53-s + 2.04·61-s + 1/8·64-s − 0.242·68-s − 0.117·72-s + 0.468·73-s − 1.85·74-s + 1/9·81-s − 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108981159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108981159\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.629005176149752438633105274587, −9.361746587069330588975575945669, −8.584868551115419667042947592500, −8.367508603382134314146293072792, −7.57323216500839527256914149425, −7.50641789037179894281138679970, −6.62762171992257665130856519803, −5.96956680213085492365415051586, −5.95967651264422152670977788910, −4.94376122657450226002634205632, −4.05372696693836488551925909574, −3.84609690192901140186989283204, −2.71372337045158087575655535987, −2.04126125287275460099535053657, −0.943136411131521382953657369703,
0.943136411131521382953657369703, 2.04126125287275460099535053657, 2.71372337045158087575655535987, 3.84609690192901140186989283204, 4.05372696693836488551925909574, 4.94376122657450226002634205632, 5.95967651264422152670977788910, 5.96956680213085492365415051586, 6.62762171992257665130856519803, 7.50641789037179894281138679970, 7.57323216500839527256914149425, 8.367508603382134314146293072792, 8.584868551115419667042947592500, 9.361746587069330588975575945669, 9.629005176149752438633105274587
