| L(s) = 1 | + 8·2-s + 40·4-s − 14·7-s + 160·8-s − 5·9-s − 22·13-s − 112·14-s + 560·16-s − 40·18-s − 176·26-s − 560·28-s − 748·29-s + 1.79e3·32-s − 200·36-s + 26·37-s + 930·47-s − 1.14e3·49-s − 880·52-s − 2.24e3·56-s − 5.98e3·58-s − 564·61-s + 70·63-s + 5.37e3·64-s − 188·67-s − 800·72-s + 1.90e3·73-s + 208·74-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.755·7-s + 7.07·8-s − 0.185·9-s − 0.469·13-s − 2.13·14-s + 35/4·16-s − 0.523·18-s − 1.32·26-s − 3.77·28-s − 4.78·29-s + 9.89·32-s − 0.925·36-s + 0.115·37-s + 2.88·47-s − 3.32·49-s − 2.34·52-s − 5.34·56-s − 13.5·58-s − 1.18·61-s + 0.139·63-s + 21/2·64-s − 0.342·67-s − 1.30·72-s + 3.04·73-s + 0.326·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5937796210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5937796210\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 22 T - 70 p T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 976 T^{4} + 5 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + p T + 92 p T^{2} + p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1256 T^{2} + 3304734 T^{4} - 1256 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16855 T^{2} + 119068688 T^{4} - 16855 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4612 T^{2} - 13083402 T^{4} - 4612 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 33016 T^{2} + 516484142 T^{4} - 33016 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 374 T + 73114 T^{2} + 374 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34680 T^{2} + 2039205262 T^{4} - 34680 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 13 T + 85670 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 118372 T^{2} + 7869297446 T^{4} - 118372 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 226235 T^{2} + 23812117248 T^{4} - 226235 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 465 T + 257308 T^{2} - 465 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 579596 T^{2} + 128307103350 T^{4} - 579596 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9452 p T^{2} + 152542023638 T^{4} - 9452 p^{7} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 282 T + 359050 T^{2} + 282 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 94 T + 468110 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1209119 T^{2} + 621074815476 T^{4} - 1209119 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 950 T + 868034 T^{2} - 950 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 722 T + 819326 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 1252 T + 1115338 T^{2} + 1252 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 507200 T^{2} + 69327294 p^{2} T^{4} - 507200 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 564 T + 959618 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.10526014301975315640986570887, −6.98651522968059610978724069009, −6.50661294930309283932843225716, −6.26028198835103211768456874552, −5.97623471507338324396192043382, −5.92073767796503480869341500305, −5.90067234647622535367912368407, −5.31800531486148730659496809036, −5.21881729332446508242618899851, −5.12253987161740481604323402077, −4.59115907300278877661107721241, −4.56182016129959240502718909096, −4.25831370335797555733243327795, −3.67627916562827261018900685637, −3.59759831202512961734570912004, −3.54616009559168051567435291721, −3.50320644375202537546570976028, −2.75865996329857221705755013410, −2.55361095365155195822025691355, −2.42530707752633157973594592676, −1.98141423169620214270764080015, −1.56303642143065045613103095468, −1.51854451639727505519517542249, −0.74466096170981417669897000150, −0.04799071306121021337238811832,
0.04799071306121021337238811832, 0.74466096170981417669897000150, 1.51854451639727505519517542249, 1.56303642143065045613103095468, 1.98141423169620214270764080015, 2.42530707752633157973594592676, 2.55361095365155195822025691355, 2.75865996329857221705755013410, 3.50320644375202537546570976028, 3.54616009559168051567435291721, 3.59759831202512961734570912004, 3.67627916562827261018900685637, 4.25831370335797555733243327795, 4.56182016129959240502718909096, 4.59115907300278877661107721241, 5.12253987161740481604323402077, 5.21881729332446508242618899851, 5.31800531486148730659496809036, 5.90067234647622535367912368407, 5.92073767796503480869341500305, 5.97623471507338324396192043382, 6.26028198835103211768456874552, 6.50661294930309283932843225716, 6.98651522968059610978724069009, 7.10526014301975315640986570887
