| L(s) = 1 | − 2·3-s − 2·5-s + 9-s − 2·11-s + 16·13-s + 4·15-s − 10·17-s + 2·19-s − 6·23-s + 25-s + 2·27-s + 4·29-s + 6·31-s + 4·33-s − 4·37-s − 32·39-s + 4·41-s + 8·43-s − 2·45-s + 4·47-s − 11·49-s + 20·51-s − 4·53-s + 4·55-s − 4·57-s + 10·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 4.43·13-s + 1.03·15-s − 2.42·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.384·27-s + 0.742·29-s + 1.07·31-s + 0.696·33-s − 0.657·37-s − 5.12·39-s + 0.624·41-s + 1.21·43-s − 0.298·45-s + 0.583·47-s − 1.57·49-s + 2.80·51-s − 0.549·53-s + 0.539·55-s − 0.529·57-s + 1.30·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9987845776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9987845776\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 220 p T^{5} + 44 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 22 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 18 T^{3} + 1404 T^{4} + 18 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 92 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 416 p T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 49 T^{2} - 468 T^{3} - 5112 T^{4} - 468 p T^{5} + 49 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 472 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 594 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 828 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.76575841889937934428922247720, −6.18983717311986688547857813221, −6.13897844993060933111956763557, −6.01032035119178788480591598129, −5.93357666909907341544406744178, −5.90872490957618443387837097946, −5.50338409206048465541359940455, −5.02445757658769768998337860414, −4.86626170238739385354311865351, −4.71419448196670543131957195938, −4.51642273204914335442525004002, −4.18536572105183433001817553627, −3.88762038522176469886057317073, −3.87309090740771047962441608634, −3.76162379294957265087118159629, −3.23043317096804768331047393302, −2.99368755516388901742215728041, −2.93142710510783801764372497745, −2.42795099006746664575324357886, −2.10609265633179177477045624820, −1.62670768274389262242642488895, −1.54522863691889775455010546698, −1.03823794449243299381288736200, −0.75560813384656464123024486625, −0.26624551051694073365444105286,
0.26624551051694073365444105286, 0.75560813384656464123024486625, 1.03823794449243299381288736200, 1.54522863691889775455010546698, 1.62670768274389262242642488895, 2.10609265633179177477045624820, 2.42795099006746664575324357886, 2.93142710510783801764372497745, 2.99368755516388901742215728041, 3.23043317096804768331047393302, 3.76162379294957265087118159629, 3.87309090740771047962441608634, 3.88762038522176469886057317073, 4.18536572105183433001817553627, 4.51642273204914335442525004002, 4.71419448196670543131957195938, 4.86626170238739385354311865351, 5.02445757658769768998337860414, 5.50338409206048465541359940455, 5.90872490957618443387837097946, 5.93357666909907341544406744178, 6.01032035119178788480591598129, 6.13897844993060933111956763557, 6.18983717311986688547857813221, 6.76575841889937934428922247720
