| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 4·13-s − 8·14-s + 5·16-s + 2·19-s − 8·26-s − 12·28-s + 4·31-s + 6·32-s − 4·37-s + 4·38-s − 4·43-s − 2·49-s − 12·52-s − 12·53-s − 16·56-s + 4·61-s + 8·62-s + 7·64-s − 16·67-s − 4·73-s − 8·74-s + 6·76-s + 4·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 1.10·13-s − 2.13·14-s + 5/4·16-s + 0.458·19-s − 1.56·26-s − 2.26·28-s + 0.718·31-s + 1.06·32-s − 0.657·37-s + 0.648·38-s − 0.609·43-s − 2/7·49-s − 1.66·52-s − 1.64·53-s − 2.13·56-s + 0.512·61-s + 1.01·62-s + 7/8·64-s − 1.95·67-s − 0.468·73-s − 0.929·74-s + 0.688·76-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.42827890411515433264187551021, −7.10919783261138359579092708112, −6.80580520404008461037141790375, −6.52491785595371188323623664064, −6.11006927922105614832268231763, −6.06051260545352622197328154707, −5.36524180923671289793688489375, −5.22960932950077418406640748047, −4.67721492299394773183167644550, −4.66338194636155862444136035613, −3.89564860960273519275571740348, −3.79355622689864536053840083075, −3.14254415834998264270871275610, −3.11210517974203165509137301249, −2.48286079645468487884422033925, −2.44406911474888411034033916247, −1.40511420006487770205224781131, −1.37796362315432232441698483175, 0, 0,
1.37796362315432232441698483175, 1.40511420006487770205224781131, 2.44406911474888411034033916247, 2.48286079645468487884422033925, 3.11210517974203165509137301249, 3.14254415834998264270871275610, 3.79355622689864536053840083075, 3.89564860960273519275571740348, 4.66338194636155862444136035613, 4.67721492299394773183167644550, 5.22960932950077418406640748047, 5.36524180923671289793688489375, 6.06051260545352622197328154707, 6.11006927922105614832268231763, 6.52491785595371188323623664064, 6.80580520404008461037141790375, 7.10919783261138359579092708112, 7.42827890411515433264187551021
