| L(s) = 1 | − 4·3-s + 3·4-s − 12·7-s + 7·9-s − 12·12-s + 32·13-s − 7·16-s − 4·19-s + 48·21-s − 5·25-s + 8·27-s − 36·28-s − 36·31-s + 21·36-s − 32·37-s − 128·39-s + 32·43-s + 28·48-s + 10·49-s + 96·52-s + 16·57-s + 164·61-s − 84·63-s − 69·64-s + 48·67-s − 148·73-s + 20·75-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 3/4·4-s − 1.71·7-s + 7/9·9-s − 12-s + 2.46·13-s − 0.437·16-s − 0.210·19-s + 16/7·21-s − 1/5·25-s + 8/27·27-s − 9/7·28-s − 1.16·31-s + 7/12·36-s − 0.864·37-s − 3.28·39-s + 0.744·43-s + 7/12·48-s + 0.204·49-s + 1.84·52-s + 0.280·57-s + 2.68·61-s − 4/3·63-s − 1.07·64-s + 0.716·67-s − 2.02·73-s + 4/15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4791261753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4791261753\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 4 T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 222 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 558 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 702 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 558 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1998 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5598 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6942 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5598 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 138 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4958 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 166 T + p^{2} T^{2} )^{2} \) |
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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
−19.26169490422754119912058216377, −19.04285858142557514784321773720, −18.01263456560359807346814205390, −17.76606447551716060483375990777, −16.49831964637036568070713956991, −16.33663545296476791170032337065, −15.92254789929129234289959207944, −15.31183624438809672746134681529, −13.95866256681387457806397154503, −13.14136327212728335570193545447, −12.68544331486549814745919762072, −11.73076231917341402814225469725, −11.05046670259334216137570596707, −10.63652752406966828619200485829, −9.556881881402575780202953749311, −8.555588049631488199187956317696, −6.89108609505129160058815916512, −6.39278214583289871634882090431, −5.68732210018541315600141812996, −3.64213518169807278952791352950,
3.64213518169807278952791352950, 5.68732210018541315600141812996, 6.39278214583289871634882090431, 6.89108609505129160058815916512, 8.555588049631488199187956317696, 9.556881881402575780202953749311, 10.63652752406966828619200485829, 11.05046670259334216137570596707, 11.73076231917341402814225469725, 12.68544331486549814745919762072, 13.14136327212728335570193545447, 13.95866256681387457806397154503, 15.31183624438809672746134681529, 15.92254789929129234289959207944, 16.33663545296476791170032337065, 16.49831964637036568070713956991, 17.76606447551716060483375990777, 18.01263456560359807346814205390, 19.04285858142557514784321773720, 19.26169490422754119912058216377
