| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 2·7-s + 4·8-s + 4·10-s − 2·11-s + 6·13-s + 4·14-s + 5·16-s + 2·17-s + 4·19-s + 6·20-s − 4·22-s + 2·23-s + 3·25-s + 12·26-s + 6·28-s + 12·29-s − 8·31-s + 6·32-s + 4·34-s + 4·35-s − 6·37-s + 8·38-s + 8·40-s + 8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s + 1.41·8-s + 1.26·10-s − 0.603·11-s + 1.66·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s + 0.917·19-s + 1.34·20-s − 0.852·22-s + 0.417·23-s + 3/5·25-s + 2.35·26-s + 1.13·28-s + 2.22·29-s − 1.43·31-s + 1.06·32-s + 0.685·34-s + 0.676·35-s − 0.986·37-s + 1.29·38-s + 1.26·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.04751671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.04751671\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 190 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 286 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 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
−7.918248148303613953466409665413, −7.66168859234389668970808913447, −7.31885201401973075353684691487, −7.16093841394744168814278542693, −6.38211178893026106986610275858, −6.35159663180195628750397799418, −5.82033943104007380940734703321, −5.77839112101603002545706262208, −5.14689016651181565802307930929, −5.13838584779232935879511658230, −4.47638867629421961909613697971, −4.40873610093171376815225683631, −3.70129670201707167518996699307, −3.38487334257725442424994041632, −3.07685517067846681006432250320, −2.62247176332169232122297000867, −2.10647872262808169559194285955, −1.74345612692942493406017137352, −1.09453459860719871893868975335, −0.895302380005463563714144706960,
0.895302380005463563714144706960, 1.09453459860719871893868975335, 1.74345612692942493406017137352, 2.10647872262808169559194285955, 2.62247176332169232122297000867, 3.07685517067846681006432250320, 3.38487334257725442424994041632, 3.70129670201707167518996699307, 4.40873610093171376815225683631, 4.47638867629421961909613697971, 5.13838584779232935879511658230, 5.14689016651181565802307930929, 5.77839112101603002545706262208, 5.82033943104007380940734703321, 6.35159663180195628750397799418, 6.38211178893026106986610275858, 7.16093841394744168814278542693, 7.31885201401973075353684691487, 7.66168859234389668970808913447, 7.918248148303613953466409665413
