| L(s) = 1 | − 12·3-s + 72·7-s + 72·9-s − 16·16-s − 864·21-s − 324·27-s + 352·31-s + 288·37-s − 1.36e3·43-s + 192·48-s + 2.59e3·49-s + 1.73e3·61-s + 5.18e3·63-s − 72·67-s + 1.44e3·73-s + 1.86e3·81-s − 4.22e3·93-s + 864·97-s − 2.08e3·103-s − 3.45e3·111-s − 1.15e3·112-s + 140·121-s + 127-s + 1.64e4·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.88·7-s + 8/3·9-s − 1/4·16-s − 8.97·21-s − 2.30·27-s + 2.03·31-s + 1.27·37-s − 4.85·43-s + 0.577·48-s + 7.55·49-s + 3.64·61-s + 10.3·63-s − 0.131·67-s + 2.30·73-s + 23/9·81-s − 4.70·93-s + 0.904·97-s − 1.99·103-s − 2.95·111-s − 0.971·112-s + 0.105·121-s + 0.000698·127-s + 11.2·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.558728384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558728384\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 p T + 8 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 36 T + 648 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 40602238 T^{4} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 1658 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 213333122 T^{4} + p^{12} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 16022 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 88 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 144 T + 10368 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 135250 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 684 T + 233928 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 21531949342 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4164554158 T^{4} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 338330 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 36 T + 648 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 456622 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 720 T + 259200 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 62498 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 519133884626 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 1399570 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 432 T + 93312 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.997146012212603341304007585373, −8.634674466776740521671367509096, −8.280660941426078869043504202041, −8.078666475792658080466047516179, −8.053137180842113086134526278747, −7.930997238022472426808396861984, −7.33245877634090366938800759382, −6.92722634701442242107813133448, −6.60391505896555994965186472373, −6.59307259919393043973594383991, −6.20636229931517778447252182126, −5.47614578783819193485956982658, −5.44906190337764352017560499267, −5.26117797842436516105067750178, −4.93622632101794118122385628234, −4.76261852094126262885110194261, −4.44622085957618714774452639525, −4.11608708710029789526259362477, −3.67435730702672811109486621017, −2.92440290459982604910301692720, −2.19605603205448084371999481877, −1.87675095617346923935682117083, −1.48072588021687701231021780592, −0.946427295464733145784625857031, −0.53345015637070189510536628929,
0.53345015637070189510536628929, 0.946427295464733145784625857031, 1.48072588021687701231021780592, 1.87675095617346923935682117083, 2.19605603205448084371999481877, 2.92440290459982604910301692720, 3.67435730702672811109486621017, 4.11608708710029789526259362477, 4.44622085957618714774452639525, 4.76261852094126262885110194261, 4.93622632101794118122385628234, 5.26117797842436516105067750178, 5.44906190337764352017560499267, 5.47614578783819193485956982658, 6.20636229931517778447252182126, 6.59307259919393043973594383991, 6.60391505896555994965186472373, 6.92722634701442242107813133448, 7.33245877634090366938800759382, 7.930997238022472426808396861984, 8.053137180842113086134526278747, 8.078666475792658080466047516179, 8.280660941426078869043504202041, 8.634674466776740521671367509096, 8.997146012212603341304007585373
