| L(s) = 1 | + 6·3-s + 27·9-s − 180·17-s − 232·19-s − 142·25-s + 108·27-s + 108·41-s − 40·43-s − 674·49-s − 1.08e3·51-s − 1.39e3·57-s − 648·59-s + 232·67-s − 2.21e3·73-s − 852·75-s + 405·81-s + 2.30e3·83-s − 1.83e3·89-s + 380·97-s + 504·107-s − 4.42e3·113-s − 2.66e3·121-s + 648·123-s + 127-s − 240·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.56·17-s − 2.80·19-s − 1.13·25-s + 0.769·27-s + 0.411·41-s − 0.141·43-s − 1.96·49-s − 2.96·51-s − 3.23·57-s − 1.42·59-s + 0.423·67-s − 3.54·73-s − 1.31·75-s + 5/9·81-s + 3.04·83-s − 2.18·89-s + 0.397·97-s + 0.455·107-s − 3.68·113-s − 2·121-s + 0.475·123-s + 0.000698·127-s − 0.163·129-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 674 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1322 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 13534 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18722 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 31246 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 100106 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 51694 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 59182 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 123290 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 497666 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1106 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 963890 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1152 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 918 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{2} \) |
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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
−9.632362491746546507778124403516, −9.151764992696714786049703105089, −8.720333606766812396359357133398, −8.670000418983766598139344983904, −7.930961881339312961116608237921, −7.88476424408970501949913558176, −6.97373214846031584573255457879, −6.78519682203176240556782300701, −6.12970043121917535779737568448, −6.10370537264322879251803697079, −4.90280524553778473167536045135, −4.63024810956987968438540404625, −3.98815539035158916265374271508, −3.97273721641095654265563563842, −2.92108406176445060313403921270, −2.49347184378073428316414511178, −1.94609970268021127763640036683, −1.59139559536864167804093866819, 0, 0,
1.59139559536864167804093866819, 1.94609970268021127763640036683, 2.49347184378073428316414511178, 2.92108406176445060313403921270, 3.97273721641095654265563563842, 3.98815539035158916265374271508, 4.63024810956987968438540404625, 4.90280524553778473167536045135, 6.10370537264322879251803697079, 6.12970043121917535779737568448, 6.78519682203176240556782300701, 6.97373214846031584573255457879, 7.88476424408970501949913558176, 7.930961881339312961116608237921, 8.670000418983766598139344983904, 8.720333606766812396359357133398, 9.151764992696714786049703105089, 9.632362491746546507778124403516
