L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s + 9-s − 4·11-s + 2·14-s + 5·16-s + 2·18-s − 8·22-s − 8·23-s − 9·25-s + 3·28-s + 2·29-s + 6·32-s + 3·36-s + 6·37-s + 18·43-s − 12·44-s − 16·46-s − 6·49-s − 18·50-s − 4·53-s + 4·56-s + 4·58-s + 63-s + 7·64-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 1/3·9-s − 1.20·11-s + 0.534·14-s + 5/4·16-s + 0.471·18-s − 1.70·22-s − 1.66·23-s − 9/5·25-s + 0.566·28-s + 0.371·29-s + 1.06·32-s + 1/2·36-s + 0.986·37-s + 2.74·43-s − 1.80·44-s − 2.35·46-s − 6/7·49-s − 2.54·50-s − 0.549·53-s + 0.534·56-s + 0.525·58-s + 0.125·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.242085563\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.242085563\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−7.76870840810291367801112655880, −7.63591160976614872787897721858, −6.93328405831007303545779012455, −6.34044350417483302017587248899, −6.19212010736600889727212981131, −5.61497296953392139440379972755, −5.22247570276464605514867142508, −4.88979839840342254384190809437, −4.26819849092611976341256235965, −3.79639747241266837059275722803, −3.65936925581879505658218130359, −2.62694575901763759143648096333, −2.30356803383598646559796068707, −1.91397133776496952020518320831, −0.75493509755239524873748249812,
0.75493509755239524873748249812, 1.91397133776496952020518320831, 2.30356803383598646559796068707, 2.62694575901763759143648096333, 3.65936925581879505658218130359, 3.79639747241266837059275722803, 4.26819849092611976341256235965, 4.88979839840342254384190809437, 5.22247570276464605514867142508, 5.61497296953392139440379972755, 6.19212010736600889727212981131, 6.34044350417483302017587248899, 6.93328405831007303545779012455, 7.63591160976614872787897721858, 7.76870840810291367801112655880