| L(s) = 1 | − 3·3-s + 14·7-s − 3·9-s − 15·11-s − 51·17-s + 27·19-s − 42·21-s − 9·23-s + 18·27-s − 12·29-s − 21·31-s + 45·33-s + 31·37-s − 20·43-s − 75·47-s + 147·49-s + 153·51-s − 57·53-s − 81·57-s − 141·59-s − 141·61-s − 42·63-s − 49·67-s + 27·69-s − 252·71-s + 45·73-s − 210·77-s + ⋯ |
| L(s) = 1 | − 3-s + 2·7-s − 1/3·9-s − 1.36·11-s − 3·17-s + 1.42·19-s − 2·21-s − 0.391·23-s + 2/3·27-s − 0.413·29-s − 0.677·31-s + 1.36·33-s + 0.837·37-s − 0.465·43-s − 1.59·47-s + 3·49-s + 3·51-s − 1.07·53-s − 1.42·57-s − 2.38·59-s − 2.31·61-s − 2/3·63-s − 0.731·67-s + 9/23·69-s − 3.54·71-s + 0.616·73-s − 2.72·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2856040591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2856040591\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 27 T + 604 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T - 448 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 31 T - 408 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 75 T + 4084 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 57 T + 440 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 141 T + 10108 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 141 T + 10348 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 49 T - 2088 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 45 T + 6004 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 99 T + 11188 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77203314139928659560084928212, −10.30889164606294289802650981018, −9.551712626442508563308807321574, −8.969251897109009723270315925100, −8.905804246121212237653338476789, −8.184481127579734496679130315611, −7.78459397375095089021536461444, −7.56310690112796074825511529298, −7.03217599242051044928547920826, −6.21674851687892022468422661497, −6.02302629311301303435145859666, −5.46576852031566262897330453106, −4.91050384184050559143940697438, −4.60002160232112338680603805334, −4.47443511609953302678460723697, −3.29324419362551245130096556408, −2.69690188492565920934702722165, −1.94570219692609133228107018188, −1.52144631369747675148343201508, −0.19625571118695790255210641001,
0.19625571118695790255210641001, 1.52144631369747675148343201508, 1.94570219692609133228107018188, 2.69690188492565920934702722165, 3.29324419362551245130096556408, 4.47443511609953302678460723697, 4.60002160232112338680603805334, 4.91050384184050559143940697438, 5.46576852031566262897330453106, 6.02302629311301303435145859666, 6.21674851687892022468422661497, 7.03217599242051044928547920826, 7.56310690112796074825511529298, 7.78459397375095089021536461444, 8.184481127579734496679130315611, 8.905804246121212237653338476789, 8.969251897109009723270315925100, 9.551712626442508563308807321574, 10.30889164606294289802650981018, 10.77203314139928659560084928212
