| L(s) = 1 | − 2·3-s + 6·5-s + 2·9-s + 20·11-s − 18·13-s − 12·15-s − 2·17-s − 46·23-s + 11·25-s − 18·27-s + 28·31-s − 40·33-s + 66·37-s + 36·39-s + 28·41-s − 30·43-s + 12·45-s + 78·47-s + 4·51-s − 14·53-s + 120·55-s − 84·61-s − 108·65-s − 14·67-s + 92·69-s + 196·71-s − 98·73-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 6/5·5-s + 2/9·9-s + 1.81·11-s − 1.38·13-s − 4/5·15-s − 0.117·17-s − 2·23-s + 0.439·25-s − 2/3·27-s + 0.903·31-s − 1.21·33-s + 1.78·37-s + 0.923·39-s + 0.682·41-s − 0.697·43-s + 4/15·45-s + 1.65·47-s + 4/51·51-s − 0.264·53-s + 2.18·55-s − 1.37·61-s − 1.66·65-s − 0.208·67-s + 4/3·69-s + 2.76·71-s − 1.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.331747893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331747893\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1618 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T + 3042 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3826 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 98 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T + 4802 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3298 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−9.719733207515548781191916935640, −9.661532658382712577044888648366, −9.498713833805024439100402946554, −8.946642129150528838252438482899, −8.373957045225957256814300372951, −7.76572449348316978503248767732, −7.58731533477582450614894501190, −6.85530750251019680047430955732, −6.38892949242716072323059818293, −6.31496761817438579474310051508, −5.65500994595167842242624495942, −5.51631636013647044561779495542, −4.70970160915753985318585709589, −4.19957913645217588710882929415, −4.05925660394284288060450122528, −3.11354035306616046368054592248, −2.37237448144610065043351343918, −2.00581744023120878846970350891, −1.32124950372146113583617411629, −0.52229640382573448402016971258,
0.52229640382573448402016971258, 1.32124950372146113583617411629, 2.00581744023120878846970350891, 2.37237448144610065043351343918, 3.11354035306616046368054592248, 4.05925660394284288060450122528, 4.19957913645217588710882929415, 4.70970160915753985318585709589, 5.51631636013647044561779495542, 5.65500994595167842242624495942, 6.31496761817438579474310051508, 6.38892949242716072323059818293, 6.85530750251019680047430955732, 7.58731533477582450614894501190, 7.76572449348316978503248767732, 8.373957045225957256814300372951, 8.946642129150528838252438482899, 9.498713833805024439100402946554, 9.661532658382712577044888648366, 9.719733207515548781191916935640
