| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 15-s + 21-s + 23-s − 2·27-s + 2·29-s − 35-s − 2·41-s + 2·43-s + 45-s − 2·47-s − 61-s − 63-s − 67-s − 69-s + 2·81-s + 2·83-s − 2·87-s + 89-s − 101-s + 103-s + 105-s − 107-s − 109-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 15-s + 21-s + 23-s − 2·27-s + 2·29-s − 35-s − 2·41-s + 2·43-s + 45-s − 2·47-s − 61-s − 63-s − 67-s − 69-s + 2·81-s + 2·83-s − 2·87-s + 89-s − 101-s + 103-s + 105-s − 107-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8913433216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8913433216\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.660357564804864029479112483186, −9.042110131425144271902329221859, −8.932330063020365438146459018233, −8.039105865548225180694187289476, −8.033661434127012660678556910288, −7.26759956010368078125159812227, −7.01375406791075073098618533716, −6.47199940873626759762564549429, −6.33171594929233881273977755939, −6.03612914152861169754015307236, −5.51686824832046571859412270895, −4.97578023946910304256527482000, −4.91807084212799962353213149726, −4.20603978466486982444028253185, −3.76519449659918662424535426814, −2.96755242277792462143074573383, −2.96451564259766717480040014077, −1.94089375369983432011768157681, −1.61469625936993888832013987506, −0.71125800536627598171293698380,
0.71125800536627598171293698380, 1.61469625936993888832013987506, 1.94089375369983432011768157681, 2.96451564259766717480040014077, 2.96755242277792462143074573383, 3.76519449659918662424535426814, 4.20603978466486982444028253185, 4.91807084212799962353213149726, 4.97578023946910304256527482000, 5.51686824832046571859412270895, 6.03612914152861169754015307236, 6.33171594929233881273977755939, 6.47199940873626759762564549429, 7.01375406791075073098618533716, 7.26759956010368078125159812227, 8.033661434127012660678556910288, 8.039105865548225180694187289476, 8.932330063020365438146459018233, 9.042110131425144271902329221859, 9.660357564804864029479112483186
