| L(s) = 1 | − 2·7-s − 9-s − 2·11-s + 4·23-s + 6·25-s − 8·29-s + 20·37-s − 6·43-s − 3·49-s − 4·53-s + 2·63-s − 2·67-s − 4·71-s + 4·77-s + 81-s + 2·99-s + 14·107-s − 4·109-s + 16·113-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1/3·9-s − 0.603·11-s + 0.834·23-s + 6/5·25-s − 1.48·29-s + 3.28·37-s − 0.914·43-s − 3/7·49-s − 0.549·53-s + 0.251·63-s − 0.244·67-s − 0.474·71-s + 0.455·77-s + 1/9·81-s + 0.201·99-s + 1.35·107-s − 0.383·109-s + 1.50·113-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359744858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359744858\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
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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
−8.041855004930813194200074093325, −7.79296219057945756346844945464, −7.31028788171323543879070919475, −6.88211692888878813136690230204, −6.24563100734937619284717614889, −6.12454811431710469750903809749, −5.48064746702700178895262951339, −4.98798584746338998364126990503, −4.58317708447283891664486002774, −3.94384468924613375055942209338, −3.33408446042211487992170394661, −2.85530229263130407316078469510, −2.46566344584681249170429715162, −1.50690458405457766837189560317, −0.55674728726783949662610747338,
0.55674728726783949662610747338, 1.50690458405457766837189560317, 2.46566344584681249170429715162, 2.85530229263130407316078469510, 3.33408446042211487992170394661, 3.94384468924613375055942209338, 4.58317708447283891664486002774, 4.98798584746338998364126990503, 5.48064746702700178895262951339, 6.12454811431710469750903809749, 6.24563100734937619284717614889, 6.88211692888878813136690230204, 7.31028788171323543879070919475, 7.79296219057945756346844945464, 8.041855004930813194200074093325
