| L(s) = 1 | + 2·5-s − 6·7-s − 12·11-s + 13-s + 2·17-s + 8·19-s + 5·25-s + 2·29-s + 2·31-s − 12·35-s − 14·37-s − 43-s + 13·49-s + 4·53-s − 24·55-s − 8·59-s + 11·61-s + 2·65-s + 15·67-s − 6·71-s − 9·73-s + 72·77-s − 13·79-s − 28·83-s + 4·85-s − 12·89-s − 6·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.26·7-s − 3.61·11-s + 0.277·13-s + 0.485·17-s + 1.83·19-s + 25-s + 0.371·29-s + 0.359·31-s − 2.02·35-s − 2.30·37-s − 0.152·43-s + 13/7·49-s + 0.549·53-s − 3.23·55-s − 1.04·59-s + 1.40·61-s + 0.248·65-s + 1.83·67-s − 0.712·71-s − 1.05·73-s + 8.20·77-s − 1.46·79-s − 3.07·83-s + 0.433·85-s − 1.27·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4348214768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4348214768\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.981083274198676911421856097437, −8.532928050683265795602384277640, −8.443830514801603050453646626056, −7.82238903424665582494560254601, −7.39233269309537169760840267157, −7.07543669308832514395901371302, −6.88728570346914776697974970290, −6.29024350182358882230374406991, −5.65842290180056899052937424514, −5.57715049177454703771850884134, −5.44856397886939491188175788190, −4.85978345911270699019597705642, −4.40122110255866120191808017015, −3.41357512345962680548915484375, −3.17115571301512295286710814125, −3.02426399838629682618637700907, −2.57123971152385969107819401121, −2.02871854344298542827948004454, −1.11555320753126228611485908748, −0.22502981386753894468013350844,
0.22502981386753894468013350844, 1.11555320753126228611485908748, 2.02871854344298542827948004454, 2.57123971152385969107819401121, 3.02426399838629682618637700907, 3.17115571301512295286710814125, 3.41357512345962680548915484375, 4.40122110255866120191808017015, 4.85978345911270699019597705642, 5.44856397886939491188175788190, 5.57715049177454703771850884134, 5.65842290180056899052937424514, 6.29024350182358882230374406991, 6.88728570346914776697974970290, 7.07543669308832514395901371302, 7.39233269309537169760840267157, 7.82238903424665582494560254601, 8.443830514801603050453646626056, 8.532928050683265795602384277640, 8.981083274198676911421856097437
