| L(s) = 1 | − 4·4-s + 4·13-s + 12·16-s + 100·19-s + 24·31-s + 144·37-s − 116·43-s + 14·49-s − 16·52-s − 80·61-s − 32·64-s − 216·67-s + 300·73-s − 400·76-s − 400·79-s − 72·97-s + 84·103-s + 224·109-s + 252·121-s − 96·124-s + 127-s + 131-s + 137-s + 139-s − 576·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 4/13·13-s + 3/4·16-s + 5.26·19-s + 0.774·31-s + 3.89·37-s − 2.69·43-s + 2/7·49-s − 0.307·52-s − 1.31·61-s − 1/2·64-s − 3.22·67-s + 4.10·73-s − 5.26·76-s − 5.06·79-s − 0.742·97-s + 0.815·103-s + 2.05·109-s + 2.08·121-s − 0.774·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 3.89·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.347243838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347243838\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 252 T^{2} + 34511 T^{4} - 252 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 164 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 888 T^{2} + 360146 T^{4} - 888 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 50 T + 1340 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 812 T^{2} + 581375 T^{4} - 812 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2788 T^{2} + 3352695 T^{4} - 2788 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 1930 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 72 T + 3187 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6276 T^{2} + 15449174 T^{4} - 6276 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 58 T + 3839 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4272 T^{2} + 9382658 T^{4} - 4272 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8960 T^{2} + 34984034 T^{4} - 8960 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6840 T^{2} + 30991922 T^{4} - 6840 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 40 T + 5574 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 108 T + 6791 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3276 T^{2} - 12941521 T^{4} - 3276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 150 T + 16108 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 200 T + 22139 T^{2} + 200 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 18692 T^{2} + 182016950 T^{4} - 18692 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 1008 T^{2} + 115251266 T^{4} + 1008 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 36 T + 13654 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.94736324435391866261222749788, −5.71004066935077083229994570172, −5.70383137062959549506768088125, −5.33702038825462963367442946046, −5.05379025757292649809783739340, −4.92113789615464220660716288801, −4.75294653941818129774918148238, −4.63295833026652890544148803923, −4.46638618171111568646994680371, −4.06767239666744645602543826658, −3.94053561289406057936827072709, −3.49562688905961598146238108879, −3.48121417925639385033612617269, −3.13397919552196146092597453248, −3.11363633588232064318735176819, −2.90661856498026122866442891666, −2.55593140208461999822080636448, −2.44875239811546016261779518335, −1.81399263608456486329275176578, −1.60850437367254378202156285866, −1.27397938039687803448573309398, −1.11504227420132265072454992820, −0.868866533139956152135757441854, −0.70505562922466942111231115955, −0.16723679108036923546737947327,
0.16723679108036923546737947327, 0.70505562922466942111231115955, 0.868866533139956152135757441854, 1.11504227420132265072454992820, 1.27397938039687803448573309398, 1.60850437367254378202156285866, 1.81399263608456486329275176578, 2.44875239811546016261779518335, 2.55593140208461999822080636448, 2.90661856498026122866442891666, 3.11363633588232064318735176819, 3.13397919552196146092597453248, 3.48121417925639385033612617269, 3.49562688905961598146238108879, 3.94053561289406057936827072709, 4.06767239666744645602543826658, 4.46638618171111568646994680371, 4.63295833026652890544148803923, 4.75294653941818129774918148238, 4.92113789615464220660716288801, 5.05379025757292649809783739340, 5.33702038825462963367442946046, 5.70383137062959549506768088125, 5.71004066935077083229994570172, 5.94736324435391866261222749788
