| L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 4·7-s + 8-s + 3·9-s − 3·11-s + 2·13-s − 4·14-s − 16-s − 6·17-s − 3·18-s + 19-s − 8·21-s + 3·22-s + 9·23-s − 2·24-s − 2·26-s − 10·27-s + 12·29-s − 8·31-s + 6·33-s + 6·34-s − 7·37-s − 38-s − 4·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 9-s − 0.904·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.229·19-s − 1.74·21-s + 0.639·22-s + 1.87·23-s − 0.408·24-s − 0.392·26-s − 1.92·27-s + 2.22·29-s − 1.43·31-s + 1.04·33-s + 1.02·34-s − 1.15·37-s − 0.162·38-s − 0.640·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7460024841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7460024841\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−11.40518483467871710459283141680, −11.17540456799210083939143694180, −10.79811574979514652914898104687, −10.55912520901289487820423052030, −10.10720215192894240477848547989, −9.146262133486748069002595758059, −9.093523937516463673638436525316, −8.435824106080819055901854354460, −8.054028167643850306705138259183, −7.40675130525996193179552229267, −7.05308508046181813925036245386, −6.57699278845753887304651988797, −5.68243447702310916146494104126, −5.41099348820503926504808815869, −4.70720066656773359075064121070, −4.56136019622450354262722794787, −3.63828460380210441255045746307, −2.49972513861462307076670749603, −1.70705559868448230388437732704, −0.76715960847554012757227895118,
0.76715960847554012757227895118, 1.70705559868448230388437732704, 2.49972513861462307076670749603, 3.63828460380210441255045746307, 4.56136019622450354262722794787, 4.70720066656773359075064121070, 5.41099348820503926504808815869, 5.68243447702310916146494104126, 6.57699278845753887304651988797, 7.05308508046181813925036245386, 7.40675130525996193179552229267, 8.054028167643850306705138259183, 8.435824106080819055901854354460, 9.093523937516463673638436525316, 9.146262133486748069002595758059, 10.10720215192894240477848547989, 10.55912520901289487820423052030, 10.79811574979514652914898104687, 11.17540456799210083939143694180, 11.40518483467871710459283141680
