| L(s) = 1 | + 12·11-s + 10·19-s − 12·29-s + 2·31-s + 13·49-s + 12·59-s − 26·61-s + 16·79-s + 24·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 2.29·19-s − 2.22·29-s + 0.359·31-s + 13/7·49-s + 1.56·59-s − 3.32·61-s + 1.80·79-s + 2.38·101-s + 1.34·109-s + 7.81·121-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 + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-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
| \(L(1)\) |
\(\approx\) |
\(4.698427969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.698427969\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.771812430891067718861478482068, −8.629679674318943130088201720546, −7.86179720746328266290098440204, −7.51602132357373174389843400926, −7.21275665529451835934570381372, −7.01890935575847896116117761371, −6.45244928990761226475218055610, −6.17726692128178121108605732325, −5.78113002357881209574159147094, −5.50584088868085216328610878521, −4.87370293754579118656211810960, −4.45307051704416614190699098179, −4.02582808433159564164356811089, −3.65673988935779684183758624800, −3.40283852643378920439273605297, −2.97972014926175197134583432130, −1.92683348019114953411336751075, −1.82065939696385397107901668761, −1.04349612918029643659575131225, −0.810328700324261265801925985096,
0.810328700324261265801925985096, 1.04349612918029643659575131225, 1.82065939696385397107901668761, 1.92683348019114953411336751075, 2.97972014926175197134583432130, 3.40283852643378920439273605297, 3.65673988935779684183758624800, 4.02582808433159564164356811089, 4.45307051704416614190699098179, 4.87370293754579118656211810960, 5.50584088868085216328610878521, 5.78113002357881209574159147094, 6.17726692128178121108605732325, 6.45244928990761226475218055610, 7.01890935575847896116117761371, 7.21275665529451835934570381372, 7.51602132357373174389843400926, 7.86179720746328266290098440204, 8.629679674318943130088201720546, 8.771812430891067718861478482068
