| L(s) = 1 | + 3-s + 2·5-s + 3·7-s + 9-s + 2·15-s − 4·16-s + 6·17-s + 3·21-s − 7·25-s + 27-s + 6·35-s + 16·37-s − 16·41-s − 2·43-s + 2·45-s + 6·47-s − 4·48-s + 2·49-s + 6·51-s + 3·63-s + 16·67-s − 7·75-s − 8·80-s + 81-s + 8·83-s + 12·85-s + 20·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.516·15-s − 16-s + 1.45·17-s + 0.654·21-s − 7/5·25-s + 0.192·27-s + 1.01·35-s + 2.63·37-s − 2.49·41-s − 0.304·43-s + 0.298·45-s + 0.875·47-s − 0.577·48-s + 2/7·49-s + 0.840·51-s + 0.377·63-s + 1.95·67-s − 0.808·75-s − 0.894·80-s + 1/9·81-s + 0.878·83-s + 1.30·85-s + 2.11·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377949122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377949122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.474175389655031890543822962245, −8.062798749262723709584757589645, −7.69169328150615235542861365765, −7.42319981131404835915630105083, −6.62515320796494421185519580492, −6.24129229481134922339259419985, −5.77387830407591933729876896929, −5.02993077790447449830764321556, −5.00201077729278538472829527671, −4.13743252782889985123227733059, −3.69516813073466781606782849555, −2.97415900422081075234427018347, −2.13307257724406321246824925475, −1.95060492849771153274411757294, −1.00822460156327212640900720126,
1.00822460156327212640900720126, 1.95060492849771153274411757294, 2.13307257724406321246824925475, 2.97415900422081075234427018347, 3.69516813073466781606782849555, 4.13743252782889985123227733059, 5.00201077729278538472829527671, 5.02993077790447449830764321556, 5.77387830407591933729876896929, 6.24129229481134922339259419985, 6.62515320796494421185519580492, 7.42319981131404835915630105083, 7.69169328150615235542861365765, 8.062798749262723709584757589645, 8.474175389655031890543822962245
