| L(s) = 1 | − 4·5-s + 4·7-s + 2·9-s + 4·13-s + 4·17-s − 12·23-s + 2·25-s − 8·29-s − 16·35-s − 12·37-s − 4·43-s − 8·45-s − 20·47-s + 8·49-s − 12·53-s + 16·59-s + 24·61-s + 8·63-s − 16·65-s − 12·67-s + 4·73-s − 5·81-s + 4·83-s − 16·85-s + 40·89-s + 16·91-s + 4·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s + 2/3·9-s + 1.10·13-s + 0.970·17-s − 2.50·23-s + 2/5·25-s − 1.48·29-s − 2.70·35-s − 1.97·37-s − 0.609·43-s − 1.19·45-s − 2.91·47-s + 8/7·49-s − 1.64·53-s + 2.08·59-s + 3.07·61-s + 1.00·63-s − 1.98·65-s − 1.46·67-s + 0.468·73-s − 5/9·81-s + 0.439·83-s − 1.73·85-s + 4.23·89-s + 1.67·91-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219199350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219199350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 4 T^{3} - 194 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2542 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 468 T^{3} + 3038 T^{4} + 468 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 164 T^{3} + 3358 T^{4} + 164 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1860 T^{3} + 15182 T^{4} + 1860 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} - 180 T^{3} - 6274 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 196 T^{3} + 3646 T^{4} - 196 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.870334763244681833547895305383, −8.523482516363023760874781360212, −8.046532119129929370378495464367, −7.997244195133179753957741507264, −7.978234756586393929514733428879, −7.70659754796252696284891729076, −7.39269586426901635017273492407, −7.20766969216210668848834724070, −6.71771889078974691931238367057, −6.36679599390417215470791497846, −6.35505972097129652685468766393, −5.75181553586140405454264746128, −5.50555446597866406431859449225, −5.30956871714691693882438195892, −4.76564576655909342336346123859, −4.66252222966729972445850437089, −4.33397540901941998130845560873, −3.80945201115467149013523338902, −3.66441629165746133048714740147, −3.40156015935742667575846942575, −3.35895523252105777061499125456, −2.03791085785182109504312541004, −1.96479209301956003668767726562, −1.68602546831422200502715032942, −0.61869269536148123988064184944,
0.61869269536148123988064184944, 1.68602546831422200502715032942, 1.96479209301956003668767726562, 2.03791085785182109504312541004, 3.35895523252105777061499125456, 3.40156015935742667575846942575, 3.66441629165746133048714740147, 3.80945201115467149013523338902, 4.33397540901941998130845560873, 4.66252222966729972445850437089, 4.76564576655909342336346123859, 5.30956871714691693882438195892, 5.50555446597866406431859449225, 5.75181553586140405454264746128, 6.35505972097129652685468766393, 6.36679599390417215470791497846, 6.71771889078974691931238367057, 7.20766969216210668848834724070, 7.39269586426901635017273492407, 7.70659754796252696284891729076, 7.978234756586393929514733428879, 7.997244195133179753957741507264, 8.046532119129929370378495464367, 8.523482516363023760874781360212, 8.870334763244681833547895305383
