| L(s) = 1 | − 2-s − 3·5-s + 7-s + 8-s + 3·10-s + 6·11-s − 13-s − 14-s − 16-s + 6·17-s + 4·19-s − 6·22-s + 5·25-s + 26-s + 6·29-s + 4·31-s − 6·34-s − 3·35-s − 14·37-s − 4·38-s − 3·40-s + 43-s + 3·47-s + 7·49-s − 5·50-s − 18·55-s + 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 1.80·11-s − 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.27·22-s + 25-s + 0.196·26-s + 1.11·29-s + 0.718·31-s − 1.02·34-s − 0.507·35-s − 2.30·37-s − 0.648·38-s − 0.474·40-s + 0.152·43-s + 0.437·47-s + 49-s − 0.707·50-s − 2.42·55-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.345092341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345092341\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.462279417180747473478856303911, −8.750522579260524808751623276279, −8.652629167678642986276071030485, −8.155889650143300288587153554201, −7.62549428258090011742442878394, −7.58037448037477428082984804782, −7.17850945404415015041854043292, −6.53117745871754265833087060861, −6.46163243316628601439587280858, −5.73363208443758482826788879221, −5.14673698695617610133305378831, −4.95968628888575496035496313293, −4.20722828422650329334961043772, −4.08880389106126506220649867300, −3.42475998419603825375071362981, −3.22921395758859085335288971148, −2.50994507194685297682424288031, −1.45256340996681945797555203739, −1.29900916600629197374557757685, −0.54067676372901253582954523560,
0.54067676372901253582954523560, 1.29900916600629197374557757685, 1.45256340996681945797555203739, 2.50994507194685297682424288031, 3.22921395758859085335288971148, 3.42475998419603825375071362981, 4.08880389106126506220649867300, 4.20722828422650329334961043772, 4.95968628888575496035496313293, 5.14673698695617610133305378831, 5.73363208443758482826788879221, 6.46163243316628601439587280858, 6.53117745871754265833087060861, 7.17850945404415015041854043292, 7.58037448037477428082984804782, 7.62549428258090011742442878394, 8.155889650143300288587153554201, 8.652629167678642986276071030485, 8.750522579260524808751623276279, 9.462279417180747473478856303911
