| L(s) = 1 | − 3·2-s + 4·4-s − 9·8-s − 6·11-s + 27·16-s + 18·22-s + 18·23-s − 25·25-s + 108·29-s − 36·32-s + 38·37-s + 116·43-s − 24·44-s − 54·46-s + 75·50-s − 6·53-s − 324·58-s + 17·64-s + 118·67-s − 228·71-s − 114·74-s + 94·79-s − 348·86-s + 54·88-s + 72·92-s − 100·100-s + 18·106-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 4-s − 9/8·8-s − 0.545·11-s + 1.68·16-s + 9/11·22-s + 0.782·23-s − 25-s + 3.72·29-s − 9/8·32-s + 1.02·37-s + 2.69·43-s − 0.545·44-s − 1.17·46-s + 3/2·50-s − 0.113·53-s − 5.58·58-s + 0.265·64-s + 1.76·67-s − 3.21·71-s − 1.54·74-s + 1.18·79-s − 4.04·86-s + 0.613·88-s + 0.782·92-s − 100-s + 9/53·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8678550848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8678550848\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 18 T - 205 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T + 75 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 2773 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T + 9435 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T + 2595 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−10.78308690273359565070625651617, −10.68076457935393661904461316811, −9.983374309912937793104415167769, −9.875606216080729254752341440185, −9.318035738626394289565019774845, −8.851002136392835035261185620507, −8.388468321560520725548666994418, −8.255952622007921359804199092635, −7.43304083886826174221837370352, −7.36223140888940720788131047715, −6.42975837145408346135198243821, −6.12725094603471405652828088756, −5.67705292592179500169925930707, −4.76075897261625717471516327652, −4.45563410306279271659710877642, −3.45790860078899279299599345519, −2.77184277996232839763501060195, −2.39169916739167919102443293781, −1.09449812271025034645981478057, −0.61664026032652470445278765570,
0.61664026032652470445278765570, 1.09449812271025034645981478057, 2.39169916739167919102443293781, 2.77184277996232839763501060195, 3.45790860078899279299599345519, 4.45563410306279271659710877642, 4.76075897261625717471516327652, 5.67705292592179500169925930707, 6.12725094603471405652828088756, 6.42975837145408346135198243821, 7.36223140888940720788131047715, 7.43304083886826174221837370352, 8.255952622007921359804199092635, 8.388468321560520725548666994418, 8.851002136392835035261185620507, 9.318035738626394289565019774845, 9.875606216080729254752341440185, 9.983374309912937793104415167769, 10.68076457935393661904461316811, 10.78308690273359565070625651617
