| L(s) = 1 | − 2·2-s + 2·4-s − 8·7-s − 4·8-s + 6·9-s + 12·13-s + 16·14-s + 8·16-s + 12·17-s − 12·18-s + 24·19-s − 8·23-s − 24·26-s − 16·28-s − 4·29-s + 24·31-s − 8·32-s − 24·34-s + 12·36-s − 48·38-s + 8·43-s + 16·46-s + 34·49-s + 24·52-s + 32·56-s + 8·58-s − 48·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 3.02·7-s − 1.41·8-s + 2·9-s + 3.32·13-s + 4.27·14-s + 2·16-s + 2.91·17-s − 2.82·18-s + 5.50·19-s − 1.66·23-s − 4.70·26-s − 3.02·28-s − 0.742·29-s + 4.31·31-s − 1.41·32-s − 4.11·34-s + 2·36-s − 7.78·38-s + 1.21·43-s + 2.35·46-s + 34/7·49-s + 3.32·52-s + 4.27·56-s + 1.05·58-s − 6.09·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.249423436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.249423436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1686 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.50717036469251850778516566632, −7.33258857322377907532285748076, −7.20435056550906796062780288895, −6.74613318462833870797045210224, −6.65820620006441924436404157661, −6.27015957438286718763420446786, −6.23376857038188056283950389357, −6.04807680005596764115924393353, −5.68414868100359874747664237165, −5.49898071924011538919163439393, −5.43725913550939024241660400801, −5.05106470830653944492648637759, −4.47835105676622924650221636379, −3.94604705228368635446209077612, −3.93006025932532165103262718221, −3.67436865630554106168039092403, −3.34347626060032032923744312738, −3.10428146471654188933183644077, −3.01975744552600554714090710833, −2.95987339637686428330878915795, −2.17868163415902152444110078863, −1.27501318462483227423971621274, −1.12423703232883803178104536502, −0.962026705807218063465035167858, −0.885077153239682944842106092550,
0.885077153239682944842106092550, 0.962026705807218063465035167858, 1.12423703232883803178104536502, 1.27501318462483227423971621274, 2.17868163415902152444110078863, 2.95987339637686428330878915795, 3.01975744552600554714090710833, 3.10428146471654188933183644077, 3.34347626060032032923744312738, 3.67436865630554106168039092403, 3.93006025932532165103262718221, 3.94604705228368635446209077612, 4.47835105676622924650221636379, 5.05106470830653944492648637759, 5.43725913550939024241660400801, 5.49898071924011538919163439393, 5.68414868100359874747664237165, 6.04807680005596764115924393353, 6.23376857038188056283950389357, 6.27015957438286718763420446786, 6.65820620006441924436404157661, 6.74613318462833870797045210224, 7.20435056550906796062780288895, 7.33258857322377907532285748076, 7.50717036469251850778516566632
