| L(s) = 1 | − 8·11-s + 8·19-s − 4·29-s − 20·41-s + 14·49-s + 8·59-s − 4·61-s − 16·71-s − 12·89-s − 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 1.83·19-s − 0.742·29-s − 3.12·41-s + 2·49-s + 1.04·59-s − 0.512·61-s − 1.89·71-s − 1.27·89-s − 1.19·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6794637951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6794637951\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.605263559349364925406938864088, −8.240739068160424536971687146811, −8.107878561342348611722039732474, −7.53610903091135741509242229825, −7.36104585214459524223835884086, −6.99870167363692156732496169856, −6.65576877527785585605330536344, −5.97240757042087996830886221946, −5.50802179409769211507318093843, −5.29130737378168045833724053957, −5.27661194791581842676425104503, −4.59587738531066745298987379298, −4.15589952131414918086272768215, −3.53534550819112318796205071521, −3.21389211045872759233315911795, −2.71208811900814005306627626976, −2.46023038652169434928824232353, −1.72229564738803625744369464785, −1.20218795794556246542650298518, −0.25067138604827700048997289124,
0.25067138604827700048997289124, 1.20218795794556246542650298518, 1.72229564738803625744369464785, 2.46023038652169434928824232353, 2.71208811900814005306627626976, 3.21389211045872759233315911795, 3.53534550819112318796205071521, 4.15589952131414918086272768215, 4.59587738531066745298987379298, 5.27661194791581842676425104503, 5.29130737378168045833724053957, 5.50802179409769211507318093843, 5.97240757042087996830886221946, 6.65576877527785585605330536344, 6.99870167363692156732496169856, 7.36104585214459524223835884086, 7.53610903091135741509242229825, 8.107878561342348611722039732474, 8.240739068160424536971687146811, 8.605263559349364925406938864088
