| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 8·10-s − 6·11-s + 6·12-s − 4·14-s − 8·15-s + 5·16-s − 8·17-s − 6·18-s − 6·19-s − 12·20-s + 4·21-s + 12·22-s − 2·23-s − 8·24-s + 5·25-s + 4·27-s + 6·28-s − 2·29-s + 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 2.52·10-s − 1.80·11-s + 1.73·12-s − 1.06·14-s − 2.06·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 1.37·19-s − 2.68·20-s + 0.872·21-s + 2.55·22-s − 0.417·23-s − 1.63·24-s + 25-s + 0.769·27-s + 1.13·28-s − 0.371·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 3 p T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 207 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.623561960004761021141007876597, −9.157654333440949697568580376054, −8.646423186529874188539171884196, −8.512179029874987613541000644532, −8.117228014943947903659384821385, −7.912297246261237378647756496310, −7.45382645052172311008444645214, −7.24874596078998473462173140795, −6.51517238425966197315864258908, −6.32910329425655374720158433612, −5.25550663995794891926432539900, −4.86472757175094707941348771755, −4.18754372725843729989254314536, −4.02501348995932098996250668558, −3.04720810223662167170504223271, −2.86416185658516695923656538755, −1.95201350757735839940759885672, −1.77945664350906640093555396519, 0, 0,
1.77945664350906640093555396519, 1.95201350757735839940759885672, 2.86416185658516695923656538755, 3.04720810223662167170504223271, 4.02501348995932098996250668558, 4.18754372725843729989254314536, 4.86472757175094707941348771755, 5.25550663995794891926432539900, 6.32910329425655374720158433612, 6.51517238425966197315864258908, 7.24874596078998473462173140795, 7.45382645052172311008444645214, 7.912297246261237378647756496310, 8.117228014943947903659384821385, 8.512179029874987613541000644532, 8.646423186529874188539171884196, 9.157654333440949697568580376054, 9.623561960004761021141007876597
