| L(s) = 1 | − 7-s + 6·11-s − 3·13-s − 3·17-s − 5·19-s − 6·23-s − 5·25-s + 3·29-s + 2·31-s − 7·37-s + 3·41-s + 3·43-s + 18·47-s − 6·49-s − 9·53-s + 30·59-s + 3·73-s − 6·77-s + 9·83-s − 3·89-s + 3·91-s − 3·97-s + 24·101-s + 8·103-s + 3·107-s − 11·109-s − 9·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.832·13-s − 0.727·17-s − 1.14·19-s − 1.25·23-s − 25-s + 0.557·29-s + 0.359·31-s − 1.15·37-s + 0.468·41-s + 0.457·43-s + 2.62·47-s − 6/7·49-s − 1.23·53-s + 3.90·59-s + 0.351·73-s − 0.683·77-s + 0.987·83-s − 0.317·89-s + 0.314·91-s − 0.304·97-s + 2.38·101-s + 0.788·103-s + 0.290·107-s − 1.05·109-s − 0.846·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678614803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678614803\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−8.863303641710569400758869881855, −8.766564795737429063828239305259, −8.072806085652691647947362662841, −7.927817999045557620103650863461, −7.30521783709530839709076233291, −6.91250975668837177835250841705, −6.72140558858466756734405577353, −6.29202745347741459106146260346, −5.93355277986632314330356637683, −5.64353420449175431426600555651, −4.93375262417516206642840198396, −4.61268337361711598924353731833, −4.02081157878349148111615025652, −3.90491175832975606772390203475, −3.55682627541920020372288068575, −2.70504336890518614108390483869, −2.15660099687245580252773285461, −2.05660924737912003910928759973, −1.16337622353428264424480523494, −0.43129282618097889744324184456,
0.43129282618097889744324184456, 1.16337622353428264424480523494, 2.05660924737912003910928759973, 2.15660099687245580252773285461, 2.70504336890518614108390483869, 3.55682627541920020372288068575, 3.90491175832975606772390203475, 4.02081157878349148111615025652, 4.61268337361711598924353731833, 4.93375262417516206642840198396, 5.64353420449175431426600555651, 5.93355277986632314330356637683, 6.29202745347741459106146260346, 6.72140558858466756734405577353, 6.91250975668837177835250841705, 7.30521783709530839709076233291, 7.927817999045557620103650863461, 8.072806085652691647947362662841, 8.766564795737429063828239305259, 8.863303641710569400758869881855
