| L(s) = 1 | − 2·3-s + 4·4-s + 3·9-s − 8·12-s + 12·16-s − 12·23-s − 2·25-s − 4·27-s + 12·29-s + 12·36-s − 2·43-s − 24·48-s + 11·49-s + 24·53-s + 2·61-s + 32·64-s + 24·69-s + 4·75-s − 22·79-s + 5·81-s − 24·87-s − 48·92-s − 8·100-s − 36·101-s − 2·103-s − 12·107-s − 16·108-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2·4-s + 9-s − 2.30·12-s + 3·16-s − 2.50·23-s − 2/5·25-s − 0.769·27-s + 2.22·29-s + 2·36-s − 0.304·43-s − 3.46·48-s + 11/7·49-s + 3.29·53-s + 0.256·61-s + 4·64-s + 2.88·69-s + 0.461·75-s − 2.47·79-s + 5/9·81-s − 2.57·87-s − 5.00·92-s − 4/5·100-s − 3.58·101-s − 0.197·103-s − 1.16·107-s − 1.53·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042959419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042959419\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.99022168541201805180365368170, −10.88459825257715122944784025758, −10.25079254161265134680590827031, −9.983196001486525948829947256632, −9.912383991182912542351654698313, −8.768005058525067394405677337667, −8.229261947696721318114790512853, −7.967477421025086437974286660438, −7.21821716447865331345990610727, −6.95468585242479492883352039003, −6.62915330582822778215013936478, −5.86167570820491382439076694934, −5.83579902619251598800908848119, −5.37784818348028851549801574913, −4.26936580159281048467713232171, −4.05798604057711294781276368355, −3.07939552385925686329295464339, −2.41577920338475777288576920701, −1.86463488811903134360907728775, −0.936485251864389765306096879002,
0.936485251864389765306096879002, 1.86463488811903134360907728775, 2.41577920338475777288576920701, 3.07939552385925686329295464339, 4.05798604057711294781276368355, 4.26936580159281048467713232171, 5.37784818348028851549801574913, 5.83579902619251598800908848119, 5.86167570820491382439076694934, 6.62915330582822778215013936478, 6.95468585242479492883352039003, 7.21821716447865331345990610727, 7.967477421025086437974286660438, 8.229261947696721318114790512853, 8.768005058525067394405677337667, 9.912383991182912542351654698313, 9.983196001486525948829947256632, 10.25079254161265134680590827031, 10.88459825257715122944784025758, 10.99022168541201805180365368170
