| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s + 7-s − 4·8-s + 8·10-s − 11-s + 13-s − 2·14-s + 5·16-s − 3·17-s − 3·19-s − 12·20-s + 2·22-s − 5·23-s + 2·25-s − 2·26-s + 3·28-s − 6·29-s − 10·31-s − 6·32-s + 6·34-s − 4·35-s − 2·37-s + 6·38-s + 16·40-s − 14·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s + 0.377·7-s − 1.41·8-s + 2.52·10-s − 0.301·11-s + 0.277·13-s − 0.534·14-s + 5/4·16-s − 0.727·17-s − 0.688·19-s − 2.68·20-s + 0.426·22-s − 1.04·23-s + 2/5·25-s − 0.392·26-s + 0.566·28-s − 1.11·29-s − 1.79·31-s − 1.06·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s + 0.973·38-s + 2.52·40-s − 2.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 144 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−10.13584358345770401042953234905, −10.06914809806822753310533686714, −9.217930316328139086163603069016, −8.975123713168213773965377021203, −8.523746805209980316711267621929, −8.095753005299843456753776851171, −7.70942579146276752016297586037, −7.59496725573606989190371983421, −6.94488942436586737620818264181, −6.55607500307732686649523299053, −5.93492862049092353964252060622, −5.39129730831884065767195333794, −4.63626967180425673112389828636, −4.13297606196451654133338922812, −3.51422167165774756982684200647, −3.20315597968137169051876135361, −1.89857937519663082123922664115, −1.81225175702583064467541270675, 0, 0,
1.81225175702583064467541270675, 1.89857937519663082123922664115, 3.20315597968137169051876135361, 3.51422167165774756982684200647, 4.13297606196451654133338922812, 4.63626967180425673112389828636, 5.39129730831884065767195333794, 5.93492862049092353964252060622, 6.55607500307732686649523299053, 6.94488942436586737620818264181, 7.59496725573606989190371983421, 7.70942579146276752016297586037, 8.095753005299843456753776851171, 8.523746805209980316711267621929, 8.975123713168213773965377021203, 9.217930316328139086163603069016, 10.06914809806822753310533686714, 10.13584358345770401042953234905
