| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 2·5-s − 4·6-s + 6·7-s − 2·8-s + 9-s − 4·10-s + 8·11-s − 2·12-s + 12·14-s + 4·15-s − 4·16-s + 4·17-s + 2·18-s − 5·19-s − 2·20-s − 12·21-s + 16·22-s − 4·23-s + 4·24-s + 25-s + 2·27-s + 6·28-s + 6·29-s + 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 2.26·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s + 2.41·11-s − 0.577·12-s + 3.20·14-s + 1.03·15-s − 16-s + 0.970·17-s + 0.471·18-s − 1.14·19-s − 0.447·20-s − 2.61·21-s + 3.41·22-s − 0.834·23-s + 0.816·24-s + 1/5·25-s + 0.384·27-s + 1.13·28-s + 1.11·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.518764515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.518764515\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 15 T^{2} + 56 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T - 9 T^{2} - 20 T^{3} + 320 T^{4} - 20 p T^{5} - 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 10 T + 34 T^{2} - 160 T^{3} - 1425 T^{4} - 160 p T^{5} + 34 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - T + 76 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 38 T^{2} + 192 T^{3} + 759 T^{4} + 192 p T^{5} - 38 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 54 T^{2} + 192 T^{3} + 2183 T^{4} + 192 p T^{5} - 54 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15 T + 45 T^{2} - 690 T^{3} + 12836 T^{4} - 690 p T^{5} + 45 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + T - 131 T^{2} - 10 T^{3} + 12312 T^{4} - 10 p T^{5} - 131 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 17 T + 208 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3 T - 59 T^{2} - 270 T^{3} - 2328 T^{4} - 270 p T^{5} - 59 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9 T - 95 T^{2} - 90 T^{3} + 17364 T^{4} - 90 p T^{5} - 95 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 10 T - 62 T^{2} - 160 T^{3} + 9423 T^{4} - 160 p T^{5} - 62 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 23 T + 316 T^{2} + 23 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.86786904527997279036503038023, −6.63015295166692057832163842987, −6.57118148296395674499647010766, −6.32406540260373116038439167142, −6.30206461455360158164205152024, −5.70646627720948714433154982051, −5.67193033445472768694989978073, −5.31059448628589767648630203124, −5.22140122930391841444508206794, −4.87861860163043832696112018399, −4.79243986977841072913787234119, −4.58427764105156740825410417130, −4.25475986192225297122924071921, −4.22453345839127926941095457048, −3.85721397265232172214004537554, −3.59068440701705431991885071646, −3.46636458552767120558125163374, −3.15068783545937049941874160077, −2.84343902936204700071077116093, −2.11963342434973698590390563303, −1.97197326711007346897190357992, −1.80742735930131534513665583303, −1.22028132867525132747550795302, −0.980964750474915473151526434784, −0.44825871053122836999705795163,
0.44825871053122836999705795163, 0.980964750474915473151526434784, 1.22028132867525132747550795302, 1.80742735930131534513665583303, 1.97197326711007346897190357992, 2.11963342434973698590390563303, 2.84343902936204700071077116093, 3.15068783545937049941874160077, 3.46636458552767120558125163374, 3.59068440701705431991885071646, 3.85721397265232172214004537554, 4.22453345839127926941095457048, 4.25475986192225297122924071921, 4.58427764105156740825410417130, 4.79243986977841072913787234119, 4.87861860163043832696112018399, 5.22140122930391841444508206794, 5.31059448628589767648630203124, 5.67193033445472768694989978073, 5.70646627720948714433154982051, 6.30206461455360158164205152024, 6.32406540260373116038439167142, 6.57118148296395674499647010766, 6.63015295166692057832163842987, 6.86786904527997279036503038023
