| L(s) = 1 | + 2-s − 5·4-s − 10·5-s − 13·7-s − 3·8-s − 10·10-s + 53·11-s − 73·13-s − 13·14-s − 29·16-s + 105·17-s + 38·19-s + 50·20-s + 53·22-s + 60·23-s + 75·25-s − 73·26-s + 65·28-s − 58·29-s − 244·31-s − 115·32-s + 105·34-s + 130·35-s − 198·37-s + 38·38-s + 30·40-s − 132·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 5/8·4-s − 0.894·5-s − 0.701·7-s − 0.132·8-s − 0.316·10-s + 1.45·11-s − 1.55·13-s − 0.248·14-s − 0.453·16-s + 1.49·17-s + 0.458·19-s + 0.559·20-s + 0.513·22-s + 0.543·23-s + 3/5·25-s − 0.550·26-s + 0.438·28-s − 0.371·29-s − 1.41·31-s − 0.635·32-s + 0.529·34-s + 0.627·35-s − 0.879·37-s + 0.162·38-s + 0.118·40-s − 0.502·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 29 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 13 T + 718 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 53 T + 3354 T^{2} - 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 73 T + 4896 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 105 T + 12572 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 p T + 13054 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 60 T + 21134 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 244 T + 72990 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 198 T + 81218 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 118582 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 276 T + 141158 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 57 T - 44194 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 824 T + 439782 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 198 T + 313918 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 722 T + 469114 T^{2} + 722 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 41 T + 553146 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 856 T + 551982 T^{2} - 856 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 786 T + 663482 T^{2} + 786 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 738 T + 1015598 T^{2} + 738 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1694 T + 1821582 T^{2} - 1694 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2291 T + 2713488 T^{2} - 2291 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1072 T + 1579806 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.124533119889656164204241330173, −8.965771157037316881647087495702, −8.020170231182580117165620230994, −7.88623690393106839681572023203, −7.48443883664329776859538917319, −6.97236808840002786756773019733, −6.69984235497220619882857576990, −6.25661225006873609350763791412, −5.51691253345004499009215574059, −5.25259269010239135856907624082, −4.67700404913916379287522566163, −4.48651233705530895346087445082, −3.66894915400576725983181682339, −3.50563128095372755353717536091, −3.17840920236228462520464121964, −2.34726874368721540521849340641, −1.59578166122678817371680223921, −1.01242874668667084489806642339, 0, 0,
1.01242874668667084489806642339, 1.59578166122678817371680223921, 2.34726874368721540521849340641, 3.17840920236228462520464121964, 3.50563128095372755353717536091, 3.66894915400576725983181682339, 4.48651233705530895346087445082, 4.67700404913916379287522566163, 5.25259269010239135856907624082, 5.51691253345004499009215574059, 6.25661225006873609350763791412, 6.69984235497220619882857576990, 6.97236808840002786756773019733, 7.48443883664329776859538917319, 7.88623690393106839681572023203, 8.020170231182580117165620230994, 8.965771157037316881647087495702, 9.124533119889656164204241330173
