| L(s) = 1 | − 2-s + 2·3-s + 2·4-s − 6·5-s − 2·6-s − 5·8-s + 3·9-s + 6·10-s − 2·11-s + 4·12-s − 2·13-s − 12·15-s + 5·16-s − 6·17-s − 3·18-s + 4·19-s − 12·20-s + 2·22-s − 2·23-s − 10·24-s + 17·25-s + 2·26-s + 4·27-s + 2·29-s + 12·30-s + 2·31-s − 10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 4-s − 2.68·5-s − 0.816·6-s − 1.76·8-s + 9-s + 1.89·10-s − 0.603·11-s + 1.15·12-s − 0.554·13-s − 3.09·15-s + 5/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 2.68·20-s + 0.426·22-s − 0.417·23-s − 2.04·24-s + 17/5·25-s + 0.392·26-s + 0.769·27-s + 0.371·29-s + 2.19·30-s + 0.359·31-s − 1.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 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
−9.051409153105574912060611094611, −8.665221783134494462776258727655, −8.364012188149483828177457165658, −7.928744199602756417395055335232, −7.75635230286110231101839373515, −7.48816643752861781008652130040, −7.02059300056720139502231850646, −6.51152250281886109482603518800, −6.36020071806973725801500989607, −5.48731017732259650576194609781, −4.77064839903175161220750390788, −4.53883250543654552372774594207, −3.97566166372297274363459789313, −3.35580587489702250586874036971, −3.24512242846668979549532372688, −2.68822293149173797406090310108, −2.22181640548165163699947672596, −1.31423005067182296472482957375, 0, 0,
1.31423005067182296472482957375, 2.22181640548165163699947672596, 2.68822293149173797406090310108, 3.24512242846668979549532372688, 3.35580587489702250586874036971, 3.97566166372297274363459789313, 4.53883250543654552372774594207, 4.77064839903175161220750390788, 5.48731017732259650576194609781, 6.36020071806973725801500989607, 6.51152250281886109482603518800, 7.02059300056720139502231850646, 7.48816643752861781008652130040, 7.75635230286110231101839373515, 7.928744199602756417395055335232, 8.364012188149483828177457165658, 8.665221783134494462776258727655, 9.051409153105574912060611094611
