| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 2·11-s − 7·13-s + 15-s + 2·17-s + 2·19-s + 21-s + 2·23-s − 6·25-s + 2·27-s − 15·29-s + 5·31-s − 2·33-s + 35-s + 16·37-s + 7·39-s − 11·41-s + 43-s + 2·45-s + 8·47-s − 10·49-s − 2·51-s + 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.603·11-s − 1.94·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.417·23-s − 6/5·25-s + 0.384·27-s − 2.78·29-s + 0.898·31-s − 0.348·33-s + 0.169·35-s + 2.63·37-s + 1.12·39-s − 1.71·41-s + 0.152·43-s + 0.298·45-s + 1.16·47-s − 1.42·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 35 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 15 T + 111 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 109 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 83 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 102 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 65 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 169 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 26 T + 302 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 167 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−8.358895437090235206759436796860, −7.83200400972601025979297758630, −7.67019693490022117680034540987, −7.50131396194742971695020905811, −6.81706128950577403993615636276, −6.74562287064492887547690602924, −5.96821453965534371336900433131, −5.88375961149779338106889285144, −5.35753755528923034961581183438, −5.18609395933032407836473225252, −4.53935269492226908953650581988, −4.19418652498217995583052543980, −3.75051428546192640425754510789, −3.33235690774399847798854618134, −2.62687982149335497551463432781, −2.53341398825740151331511385314, −1.72584894135688654187460871474, −1.07731290027739869936185854387, 0, 0,
1.07731290027739869936185854387, 1.72584894135688654187460871474, 2.53341398825740151331511385314, 2.62687982149335497551463432781, 3.33235690774399847798854618134, 3.75051428546192640425754510789, 4.19418652498217995583052543980, 4.53935269492226908953650581988, 5.18609395933032407836473225252, 5.35753755528923034961581183438, 5.88375961149779338106889285144, 5.96821453965534371336900433131, 6.74562287064492887547690602924, 6.81706128950577403993615636276, 7.50131396194742971695020905811, 7.67019693490022117680034540987, 7.83200400972601025979297758630, 8.358895437090235206759436796860
