| L(s) = 1 | + 4·2-s + 12·4-s + 6·5-s + 32·8-s + 24·10-s + 10·13-s + 80·16-s − 14·17-s + 72·20-s + 11·25-s + 40·26-s + 192·32-s − 56·34-s + 192·40-s − 62·41-s − 98·49-s + 44·50-s + 120·52-s − 34·53-s − 240·61-s + 448·64-s + 60·65-s − 168·68-s − 192·73-s + 480·80-s − 81·81-s − 248·82-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 6/5·5-s + 4·8-s + 12/5·10-s + 0.769·13-s + 5·16-s − 0.823·17-s + 18/5·20-s + 0.439·25-s + 1.53·26-s + 6·32-s − 1.64·34-s + 24/5·40-s − 1.51·41-s − 2·49-s + 0.879·50-s + 2.30·52-s − 0.641·53-s − 3.93·61-s + 7·64-s + 0.923·65-s − 2.47·68-s − 2.63·73-s + 6·80-s − 81-s − 3.02·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.62523548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.62523548\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 10 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )( 1 + 80 T + p^{2} T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )( 1 - 78 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
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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
−11.93165103260004580316781420141, −11.80202248749234617415726514968, −11.11752817767022085157805487525, −10.84259717672418925315208544445, −10.10271552361107304312090602705, −10.09723060825749274082088701882, −9.071634788893896485271876477965, −8.684196374967471558439317800067, −7.71106739610626369407121621184, −7.49930424826535043140339394333, −6.44820152988945831407863003802, −6.43138431945820709534052102071, −5.97226287723578829502369695894, −5.31531988990860388292484490362, −4.70827043213986551919886784285, −4.36678272759963411005017675459, −3.29193651152931301702177022742, −3.06484934319142439406657335536, −1.91759532515871146067610562991, −1.62519318980755664968012498080,
1.62519318980755664968012498080, 1.91759532515871146067610562991, 3.06484934319142439406657335536, 3.29193651152931301702177022742, 4.36678272759963411005017675459, 4.70827043213986551919886784285, 5.31531988990860388292484490362, 5.97226287723578829502369695894, 6.43138431945820709534052102071, 6.44820152988945831407863003802, 7.49930424826535043140339394333, 7.71106739610626369407121621184, 8.684196374967471558439317800067, 9.071634788893896485271876477965, 10.09723060825749274082088701882, 10.10271552361107304312090602705, 10.84259717672418925315208544445, 11.11752817767022085157805487525, 11.80202248749234617415726514968, 11.93165103260004580316781420141
