| L(s) = 1 | − 4·2-s + 12·4-s − 8·5-s + 12·7-s − 32·8-s + 32·10-s − 48·11-s + 84·13-s − 48·14-s + 80·16-s − 104·17-s + 28·19-s − 96·20-s + 192·22-s − 46·23-s − 74·25-s − 336·26-s + 144·28-s − 132·29-s − 8·31-s − 192·32-s + 416·34-s − 96·35-s − 324·37-s − 112·38-s + 256·40-s + 140·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.715·5-s + 0.647·7-s − 1.41·8-s + 1.01·10-s − 1.31·11-s + 1.79·13-s − 0.916·14-s + 5/4·16-s − 1.48·17-s + 0.338·19-s − 1.07·20-s + 1.86·22-s − 0.417·23-s − 0.591·25-s − 2.53·26-s + 0.971·28-s − 0.845·29-s − 0.0463·31-s − 1.06·32-s + 2.09·34-s − 0.463·35-s − 1.43·37-s − 0.478·38-s + 1.01·40-s + 0.533·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 12 T + 594 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 48 T + 1190 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 84 T + 4110 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 T + 11378 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 28 T + 12762 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 132 T + 40334 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 46798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 324 T + 65598 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 140 T + 124310 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 548 T + 218602 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 464 T + 256862 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 336 T + 110810 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 72 T - 79978 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 60 T + 184014 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 444 T + 557498 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 672 T + 427310 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 252 T + 447798 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1692 T + 1701666 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 400 T + 192342 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 336 T + 1391954 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 220 T + 976774 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.48384063040260102038804511646, −10.38736563868443191599140933450, −9.504800213098410368870686183925, −9.173597233161851789795618417567, −8.586865103708180023846535287671, −8.348867092084888479250315466615, −7.87726253472385184918461047146, −7.65751893292048251916474931611, −6.82780873290189758325318073508, −6.67091035029967032304593447353, −5.73577539882838528660692622217, −5.54454589810460713650879819794, −4.62503409436491508097606741070, −4.03047778425420720641122043156, −3.36366399835491682723759360429, −2.69751051312056910549723364037, −1.79049007682542678604872165044, −1.41049147118470010484629705013, 0, 0,
1.41049147118470010484629705013, 1.79049007682542678604872165044, 2.69751051312056910549723364037, 3.36366399835491682723759360429, 4.03047778425420720641122043156, 4.62503409436491508097606741070, 5.54454589810460713650879819794, 5.73577539882838528660692622217, 6.67091035029967032304593447353, 6.82780873290189758325318073508, 7.65751893292048251916474931611, 7.87726253472385184918461047146, 8.348867092084888479250315466615, 8.586865103708180023846535287671, 9.173597233161851789795618417567, 9.504800213098410368870686183925, 10.38736563868443191599140933450, 10.48384063040260102038804511646
