| L(s) = 1 | − 190·3-s + 512·4-s + 1.96e4·9-s + 1.98e5·11-s − 9.72e4·12-s − 1.98e6·19-s + 3.90e6·25-s − 4.36e6·27-s − 3.76e7·33-s + 1.00e7·36-s − 1.02e8·41-s − 4.47e7·43-s + 1.01e8·44-s − 8.07e7·49-s + 3.76e8·57-s − 2.52e8·59-s − 1.34e8·64-s + 6.28e7·67-s − 8.44e8·73-s − 7.42e8·75-s − 1.01e9·76-s + 8.28e8·81-s + 1.73e9·97-s + 3.89e9·99-s + 2.00e9·100-s − 2.23e9·108-s + 1.94e10·121-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 4-s + 9-s + 4.08·11-s − 1.35·12-s − 3.48·19-s + 2·25-s − 1.57·27-s − 5.52·33-s + 36-s − 5.68·41-s − 1.99·43-s + 4.08·44-s − 2·49-s + 4.72·57-s − 2.71·59-s − 64-s + 0.380·67-s − 3.48·73-s − 2.70·75-s − 3.48·76-s + 2.13·81-s + 1.99·97-s + 4.08·99-s + 2·100-s − 1.57·108-s + 8.24·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.348706488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.348706488\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 190 T + 16417 T^{2} + 190 p^{9} T^{3} + p^{18} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 66042 T + p^{9} T^{2} )^{2}( 1 - 66042 T + 2003598073 T^{2} - 66042 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 597510 T + 238430323603 T^{2} - 597510 p^{9} T^{3} + p^{18} T^{4} )( 1 + 597510 T + 238430323603 T^{2} + 597510 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 990146 T + 657701403537 T^{2} + 990146 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 34306362 T + p^{9} T^{2} )^{2}( 1 + 34306362 T + 849544539281083 T^{2} + 34306362 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 44782090 T + p^{9} T^{2} )^{2}( 1 - 44782090 T + 1502842972831257 T^{2} - 44782090 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 84243834 T + p^{9} T^{2} )^{2}( 1 + 84243834 T - 1565972251635383 T^{2} + 84243834 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 62817230 T + p^{9} T^{2} )^{2}( 1 + 62817230 T - 23260530011422047 T^{2} + 62817230 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 422324930 T + 119486759791236987 T^{2} + 422324930 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 144637650 T - 166020205470017903 T^{2} - 144637650 p^{9} T^{3} + p^{18} T^{4} )( 1 + 144637650 T - 166020205470017903 T^{2} + 144637650 p^{9} T^{3} + p^{18} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 1089849006 T + p^{9} T^{2} )^{2}( 1 + 1089849006 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1738254710 T + p^{9} T^{2} )^{2}( 1 + 1738254710 T + 2261298378182618883 T^{2} + 1738254710 p^{9} T^{3} + p^{18} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.852244164101031622849043913045, −8.689873913570018757066015990875, −8.520445178413951783484848374138, −8.033472009608271608050724404928, −7.46479745826285174952485887877, −6.91575656025677203889221997368, −6.78186356282441126473118584531, −6.69716104826998704160404927411, −6.32150793311194887857014957010, −6.28188867896929149739401661819, −6.14060409783990931991132541731, −5.32132995775059223760909785298, −4.86479121218185684519544572453, −4.70306299316123994727477925501, −4.37379279866844624093927043777, −3.90951605935995986361925723160, −3.61929201003955848077558904551, −3.21656955008122095712010123260, −2.85830396556037897679358476739, −1.72555992839170177076604042006, −1.71393997184272366286539458461, −1.70662027837813543095987527626, −1.41116600393785443991400843878, −0.48439531735700533473667157728, −0.22080519075705065120747666065,
0.22080519075705065120747666065, 0.48439531735700533473667157728, 1.41116600393785443991400843878, 1.70662027837813543095987527626, 1.71393997184272366286539458461, 1.72555992839170177076604042006, 2.85830396556037897679358476739, 3.21656955008122095712010123260, 3.61929201003955848077558904551, 3.90951605935995986361925723160, 4.37379279866844624093927043777, 4.70306299316123994727477925501, 4.86479121218185684519544572453, 5.32132995775059223760909785298, 6.14060409783990931991132541731, 6.28188867896929149739401661819, 6.32150793311194887857014957010, 6.69716104826998704160404927411, 6.78186356282441126473118584531, 6.91575656025677203889221997368, 7.46479745826285174952485887877, 8.033472009608271608050724404928, 8.520445178413951783484848374138, 8.689873913570018757066015990875, 8.852244164101031622849043913045
