| L(s) = 1 | − 836·7-s + 440·13-s + 6.68e3·19-s + 1.16e4·25-s − 1.60e5·31-s − 1.50e5·37-s − 9.01e4·43-s + 1.28e4·49-s + 5.84e5·61-s + 7.66e5·67-s + 1.57e6·73-s − 3.23e5·79-s − 3.67e5·91-s + 4.43e6·97-s + 3.36e5·103-s − 3.67e6·109-s + 4.42e6·121-s + 127-s + 131-s − 5.59e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.43·7-s + 0.200·13-s + 0.975·19-s + 0.745·25-s − 5.37·31-s − 2.97·37-s − 1.13·43-s + 0.109·49-s + 2.57·61-s + 2.54·67-s + 4.04·73-s − 0.655·79-s − 0.488·91-s + 4.85·97-s + 0.307·103-s − 2.83·109-s + 2.49·121-s − 2.37·133-s + 0.703·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1790287500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1790287500\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 466 p^{2} T^{2} - 4269 p^{4} T^{4} - 466 p^{14} T^{6} + p^{24} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 418 T + 255651 T^{2} + 418 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4420066 T^{2} + 9395406294531 T^{4} - 4420066 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 220 T - 1625034 T^{2} - 220 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 56197372 T^{2} + 1562088612242310 T^{4} - 56197372 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 176 p T + 47525298 T^{2} - 176 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 419760292 T^{2} + 80602923439210758 T^{4} - 419760292 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2090745436 T^{2} + 1792929545635300134 T^{4} - 2090745436 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 80086 T + 3373200411 T^{2} + 80086 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 75272 T + 6285807906 T^{2} + 75272 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5889502372 T^{2} + 16231333840281591558 T^{4} - 5889502372 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 45076 T + 3172928694 T^{2} + 45076 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6782031292 T^{2} - 77190653510200977402 T^{4} - 6782031292 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 28851606590 T^{2} + \)\(11\!\cdots\!19\)\( T^{4} + 28851606590 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 156087793948 T^{2} + \)\(96\!\cdots\!10\)\( T^{4} - 156087793948 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 292072 T + 122096421186 T^{2} - 292072 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 383396 T + 214539961974 T^{2} - 383396 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 273680879548 T^{2} + \)\(49\!\cdots\!90\)\( T^{4} - 273680879548 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10786 p T + 457047261267 T^{2} - 10786 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 161500 T - 89571779706 T^{2} + 161500 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 733177338322 T^{2} + \)\(34\!\cdots\!35\)\( T^{4} - 733177338322 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 995915389756 T^{2} + \)\(73\!\cdots\!54\)\( T^{4} - 995915389756 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 2216470 T + 2476328450811 T^{2} - 2216470 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−6.98454115058203418196120345567, −6.73861966658859985238859667595, −6.65094258215525429001939641122, −6.37419591638479105955530628791, −6.23303717034264189154514817134, −5.69815109697772909204339797797, −5.34119658778107164616866663450, −5.30372080328047366719926911845, −5.28228987216577357658567116011, −4.92590430104945738401078844993, −4.53167520608583536701136507837, −3.80231233952314787257948127391, −3.65422179285480368535380290218, −3.61993191909023299196437025571, −3.55786905362273386292929170718, −3.12021098837654597112256316327, −3.08848907470352392343264680968, −2.25537788690189100192199636763, −2.16757282830658270386482703719, −1.93698177915167070189229331122, −1.61823015405595671186238173630, −1.09552978688367927579509732160, −0.75535124999744212179860215311, −0.36626311687649279311886316326, −0.07773045027742502801852549046,
0.07773045027742502801852549046, 0.36626311687649279311886316326, 0.75535124999744212179860215311, 1.09552978688367927579509732160, 1.61823015405595671186238173630, 1.93698177915167070189229331122, 2.16757282830658270386482703719, 2.25537788690189100192199636763, 3.08848907470352392343264680968, 3.12021098837654597112256316327, 3.55786905362273386292929170718, 3.61993191909023299196437025571, 3.65422179285480368535380290218, 3.80231233952314787257948127391, 4.53167520608583536701136507837, 4.92590430104945738401078844993, 5.28228987216577357658567116011, 5.30372080328047366719926911845, 5.34119658778107164616866663450, 5.69815109697772909204339797797, 6.23303717034264189154514817134, 6.37419591638479105955530628791, 6.65094258215525429001939641122, 6.73861966658859985238859667595, 6.98454115058203418196120345567
