| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 6·9-s − 4·13-s + 5·16-s − 2·17-s + 12·18-s − 16·19-s − 25-s − 8·26-s + 6·32-s − 4·34-s + 18·36-s − 32·38-s − 16·43-s + 10·49-s − 2·50-s − 12·52-s + 20·53-s + 8·59-s + 7·64-s − 8·67-s − 6·68-s + 24·72-s − 48·76-s + 27·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2·9-s − 1.10·13-s + 5/4·16-s − 0.485·17-s + 2.82·18-s − 3.67·19-s − 1/5·25-s − 1.56·26-s + 1.06·32-s − 0.685·34-s + 3·36-s − 5.19·38-s − 2.43·43-s + 10/7·49-s − 0.282·50-s − 1.66·52-s + 2.74·53-s + 1.04·59-s + 7/8·64-s − 0.977·67-s − 0.727·68-s + 2.82·72-s − 5.50·76-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.878557579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878557579\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−12.94407642036537484328674743374, −12.71991391950053845994315374925, −12.19336794839424565471569783485, −11.85433668855291833341100897616, −11.05543305765230238128605601836, −10.55622503560228451325784433348, −10.11639100699523557853309705068, −9.945609295567579133963362296170, −8.719628844099429559617781278284, −8.538087315103129226148980569814, −7.37919842515154954497070386172, −7.26835572617531933661978272572, −6.45168467505020539170819963411, −6.34769481191182450507979128760, −5.21583820190124830450662575045, −4.66481553244488061897850650772, −4.16309584125763120004185199517, −3.81144439505337770209116126257, −2.35577095351504871124232407459, −1.96873713218844423806568882524,
1.96873713218844423806568882524, 2.35577095351504871124232407459, 3.81144439505337770209116126257, 4.16309584125763120004185199517, 4.66481553244488061897850650772, 5.21583820190124830450662575045, 6.34769481191182450507979128760, 6.45168467505020539170819963411, 7.26835572617531933661978272572, 7.37919842515154954497070386172, 8.538087315103129226148980569814, 8.719628844099429559617781278284, 9.945609295567579133963362296170, 10.11639100699523557853309705068, 10.55622503560228451325784433348, 11.05543305765230238128605601836, 11.85433668855291833341100897616, 12.19336794839424565471569783485, 12.71991391950053845994315374925, 12.94407642036537484328674743374
