| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·7-s − 2·8-s − 3·9-s + 3·11-s − 2·12-s + 14·13-s + 4·14-s − 4·16-s + 3·17-s − 6·18-s + 19-s − 4·21-s + 6·22-s + 9·23-s + 4·24-s + 13·25-s + 28·26-s + 14·27-s + 2·28-s + 3·29-s + 4·31-s − 2·32-s − 6·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s + 0.755·7-s − 0.707·8-s − 9-s + 0.904·11-s − 0.577·12-s + 3.88·13-s + 1.06·14-s − 16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s − 0.872·21-s + 1.27·22-s + 1.87·23-s + 0.816·24-s + 13/5·25-s + 5.49·26-s + 2.69·27-s + 0.377·28-s + 0.557·29-s + 0.718·31-s − 0.353·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.038714706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.038714706\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 29 T^{2} + 8 T^{3} + 520 T^{4} + 8 p T^{5} - 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9 T + 77 T^{2} - 450 T^{3} + 2592 T^{4} - 450 p T^{5} + 77 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 168 p T^{5} + 59 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 27 T + 383 T^{2} + 3780 T^{3} + 27882 T^{4} + 3780 p T^{5} + 383 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - T^{2} - 4488 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 27 T + 419 T^{2} + 4752 T^{3} + 41832 T^{4} + 4752 p T^{5} + 419 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 15 T + 101 T^{2} - 270 T^{3} - 5640 T^{4} - 270 p T^{5} + 101 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 25 T + 306 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 36750 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 132 T^{3} - 5718 T^{4} - 132 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 10 T - 86 T^{2} + 80 T^{3} + 15487 T^{4} + 80 p T^{5} - 86 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.73334733343092793009558938534, −7.61248548259118218785163035858, −7.00278946418370196944288660259, −6.78419804036510481825922326936, −6.62438171519103082851791572383, −6.50645569732145907421714534583, −6.37424947578286834094102646723, −5.93268575811222532677011065666, −5.86919118152622788792982332433, −5.51512702831221887789329282710, −5.40754270409293217259494689245, −4.85807504380340642625059993047, −4.81948979405628088833365284234, −4.69697389091039334831182744701, −4.64243794126639984351978408862, −3.73304620807696545768218450700, −3.61527386605545495693374086724, −3.48519190233748406800961158027, −3.26030631328793554358354183050, −2.95152889460607902624652597497, −2.72042693565101944041330524495, −1.76812145425410242318035840060, −1.48874576662674745439857809010, −1.03266414855375778945310815257, −0.78724608820131961066438979290,
0.78724608820131961066438979290, 1.03266414855375778945310815257, 1.48874576662674745439857809010, 1.76812145425410242318035840060, 2.72042693565101944041330524495, 2.95152889460607902624652597497, 3.26030631328793554358354183050, 3.48519190233748406800961158027, 3.61527386605545495693374086724, 3.73304620807696545768218450700, 4.64243794126639984351978408862, 4.69697389091039334831182744701, 4.81948979405628088833365284234, 4.85807504380340642625059993047, 5.40754270409293217259494689245, 5.51512702831221887789329282710, 5.86919118152622788792982332433, 5.93268575811222532677011065666, 6.37424947578286834094102646723, 6.50645569732145907421714534583, 6.62438171519103082851791572383, 6.78419804036510481825922326936, 7.00278946418370196944288660259, 7.61248548259118218785163035858, 7.73334733343092793009558938534
