| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 2·12-s + 4·13-s + 16-s + 18-s + 2·24-s − 25-s + 4·26-s − 4·27-s + 12·29-s − 2·31-s + 32-s + 36-s + 8·39-s − 10·43-s + 2·48-s + 5·49-s − 50-s + 4·52-s − 4·54-s + 12·58-s − 2·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s + 0.408·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s + 2.22·29-s − 0.359·31-s + 0.176·32-s + 1/6·36-s + 1.28·39-s − 1.52·43-s + 0.288·48-s + 5/7·49-s − 0.141·50-s + 0.554·52-s − 0.544·54-s + 1.57·58-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.828605526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.828605526\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.192075759695813155898070332935, −8.709677502245832111164745474146, −8.542146293378825267371721793671, −7.85595812450323828298989876530, −7.60552816872274133401913031039, −6.69127389080689754995591989925, −6.52170290801906927975113289997, −5.82651928106294987861857484876, −5.25576965010038807572292711135, −4.58633940771354719330557201522, −3.99357484974617723445642829054, −3.42128866695581214371712977106, −2.93395870739057818229643662373, −2.24324194408493933435776322455, −1.32525576890913898417929599838,
1.32525576890913898417929599838, 2.24324194408493933435776322455, 2.93395870739057818229643662373, 3.42128866695581214371712977106, 3.99357484974617723445642829054, 4.58633940771354719330557201522, 5.25576965010038807572292711135, 5.82651928106294987861857484876, 6.52170290801906927975113289997, 6.69127389080689754995591989925, 7.60552816872274133401913031039, 7.85595812450323828298989876530, 8.542146293378825267371721793671, 8.709677502245832111164745474146, 9.192075759695813155898070332935
