| L(s) = 1 | + 2·2-s + 2·4-s − 2·7-s + 2·9-s − 4·14-s − 4·16-s + 8·17-s + 4·18-s − 25-s − 4·28-s + 16·31-s − 8·32-s + 16·34-s + 4·36-s + 12·41-s + 3·49-s − 2·50-s + 32·62-s − 4·63-s − 8·64-s + 16·68-s + 20·71-s − 32·73-s − 20·79-s − 5·81-s + 24·82-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.755·7-s + 2/3·9-s − 1.06·14-s − 16-s + 1.94·17-s + 0.942·18-s − 1/5·25-s − 0.755·28-s + 2.87·31-s − 1.41·32-s + 2.74·34-s + 2/3·36-s + 1.87·41-s + 3/7·49-s − 0.282·50-s + 4.06·62-s − 0.503·63-s − 64-s + 1.94·68-s + 2.37·71-s − 3.74·73-s − 2.25·79-s − 5/9·81-s + 2.65·82-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.004775141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.004775141\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−12.41186137753275234962264906388, −11.80433142720499769100113307625, −11.42569361503958433839779207063, −10.78305304974036578035625742152, −10.03903333563229535939805958398, −9.906669157450038410347803643780, −9.521511199399975899817715929485, −8.621859705101155471355871214050, −8.287133428413997770254537132225, −7.34289588874629829334148303771, −7.28064496810896315640655674834, −6.38276865494737165617879095962, −6.01998096856598715239048300517, −5.61278445806465705484825431922, −4.86545783087627992938728755007, −4.28418177299017197253528375738, −3.85560801626718271020802458775, −2.95304584586790300065617720794, −2.72357108787803663049515683629, −1.21468062749211983890250627448,
1.21468062749211983890250627448, 2.72357108787803663049515683629, 2.95304584586790300065617720794, 3.85560801626718271020802458775, 4.28418177299017197253528375738, 4.86545783087627992938728755007, 5.61278445806465705484825431922, 6.01998096856598715239048300517, 6.38276865494737165617879095962, 7.28064496810896315640655674834, 7.34289588874629829334148303771, 8.287133428413997770254537132225, 8.621859705101155471355871214050, 9.521511199399975899817715929485, 9.906669157450038410347803643780, 10.03903333563229535939805958398, 10.78305304974036578035625742152, 11.42569361503958433839779207063, 11.80433142720499769100113307625, 12.41186137753275234962264906388
