L(s) = 1 | − 12·2-s + 72·4-s − 72·7-s − 264·8-s + 24·11-s + 144·13-s + 864·14-s + 220·16-s − 600·17-s − 288·22-s − 888·23-s − 1.72e3·26-s − 5.18e3·28-s + 3.24e3·31-s + 4.32e3·32-s + 7.20e3·34-s + 2.44e3·37-s − 6.86e3·41-s − 5.54e3·43-s + 1.72e3·44-s + 1.06e4·46-s − 5.61e3·47-s + 2.59e3·49-s + 1.03e4·52-s − 1.84e3·53-s + 1.90e4·56-s + 1.50e4·61-s + ⋯ |
L(s) = 1 | − 3·2-s + 9/2·4-s − 1.46·7-s − 4.12·8-s + 0.198·11-s + 0.852·13-s + 4.40·14-s + 0.859·16-s − 2.07·17-s − 0.595·22-s − 1.67·23-s − 2.55·26-s − 6.61·28-s + 3.37·31-s + 4.21·32-s + 6.22·34-s + 1.78·37-s − 4.08·41-s − 2.99·43-s + 0.892·44-s + 5.03·46-s − 2.54·47-s + 1.07·49-s + 3.83·52-s − 0.657·53-s + 6.06·56-s + 4.03·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4565376782\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4565376782\) |
\(L(3)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $D_4\times C_2$ | \( 1 + 3 p^{2} T + 9 p^{3} T^{2} + 33 p^{3} T^{3} + 233 p^{2} T^{4} + 33 p^{7} T^{5} + 9 p^{11} T^{6} + 3 p^{14} T^{7} + p^{16} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 219312 T^{3} + 18140207 T^{4} + 219312 p^{4} T^{5} + 2592 p^{8} T^{6} + 72 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 12 T - 7186 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 144 T + 10368 T^{2} - 2429280 T^{3} + 432514319 T^{4} - 2429280 p^{4} T^{5} + 10368 p^{8} T^{6} - 144 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 600 T + 180000 T^{2} + 77105400 T^{3} + 31005206018 T^{4} + 77105400 p^{4} T^{5} + 180000 p^{8} T^{6} + 600 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 326690 T^{2} + 54622417923 T^{4} - 326690 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 888 T + 394272 T^{2} + 52907928 T^{3} - 41414676478 T^{4} + 52907928 p^{4} T^{5} + 394272 p^{8} T^{6} + 888 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 63932 p T^{2} + 1839990957414 T^{4} - 63932 p^{9} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 1622 T + 2141667 T^{2} - 1622 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2448 T + 2996352 T^{2} - 6308792208 T^{3} + 12788952535682 T^{4} - 6308792208 p^{4} T^{5} + 2996352 p^{8} T^{6} - 2448 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3432 T + 8077562 T^{2} + 3432 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5544 T + 15367968 T^{2} + 34365692592 T^{3} + 69120271160159 T^{4} + 34365692592 p^{4} T^{5} + 15367968 p^{8} T^{6} + 5544 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 5616 T + 15769728 T^{2} + 24055647408 T^{3} + 36339718329314 T^{4} + 24055647408 p^{4} T^{5} + 15769728 p^{8} T^{6} + 5616 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 1848 T + 1707552 T^{2} + 12837022968 T^{3} + 95614873611362 T^{4} + 12837022968 p^{4} T^{5} + 1707552 p^{8} T^{6} + 1848 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10941820 T^{2} + 100216939084998 T^{4} - 10941820 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7510 T + 35401563 T^{2} - 7510 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 13320 T + 88711200 T^{2} + 439054559280 T^{3} + 2008874003467343 T^{4} + 439054559280 p^{4} T^{5} + 88711200 p^{8} T^{6} + 13320 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10776 T + 74664506 T^{2} - 10776 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 19728 T + 194596992 T^{2} - 1434198789648 T^{3} + 8607659366333762 T^{4} - 1434198789648 p^{4} T^{5} + 194596992 p^{8} T^{6} - 19728 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 25715324 T^{2} + 2139409705359366 T^{4} - 25715324 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 8592 T + 36911232 T^{2} + 464368361424 T^{3} + 5798663913908642 T^{4} + 464368361424 p^{4} T^{5} + 36911232 p^{8} T^{6} + 8592 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 203730340 T^{2} + 18051837318885318 T^{4} - 203730340 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 11520 T + 66355200 T^{2} + 1104271879680 T^{3} + 18323409093272303 T^{4} + 1104271879680 p^{4} T^{5} + 66355200 p^{8} T^{6} + 11520 p^{12} T^{7} + p^{16} T^{8} \) |
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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
−8.382362312297584403921736391029, −8.285009514425261536606862730100, −7.87812821817991840514144201866, −7.84707080294341083990727521011, −6.85236417665708870950890672487, −6.84540041012799475709303137732, −6.81753576548860598969197882829, −6.78007886727422031769963349702, −6.23263701371292065621197775728, −6.15701734234634096751959333112, −5.84864965450425930601027105907, −4.93007695270004322912043205387, −4.82522139671866812909908280537, −4.70556754561475727879793889573, −4.26770725910292131673776824166, −3.49225738788241154555190202131, −3.48728783769269585981123515822, −3.23241000854690988859096143120, −2.37686793037892801058443048257, −2.27657183225363109289877669179, −1.76204595909258146720295669151, −1.70192340428518103648228015369, −0.66934435539707814803079890515, −0.49938773129759795278055456989, −0.43918422657607800956065491415,
0.43918422657607800956065491415, 0.49938773129759795278055456989, 0.66934435539707814803079890515, 1.70192340428518103648228015369, 1.76204595909258146720295669151, 2.27657183225363109289877669179, 2.37686793037892801058443048257, 3.23241000854690988859096143120, 3.48728783769269585981123515822, 3.49225738788241154555190202131, 4.26770725910292131673776824166, 4.70556754561475727879793889573, 4.82522139671866812909908280537, 4.93007695270004322912043205387, 5.84864965450425930601027105907, 6.15701734234634096751959333112, 6.23263701371292065621197775728, 6.78007886727422031769963349702, 6.81753576548860598969197882829, 6.84540041012799475709303137732, 6.85236417665708870950890672487, 7.84707080294341083990727521011, 7.87812821817991840514144201866, 8.285009514425261536606862730100, 8.382362312297584403921736391029