| L(s) = 1 | − 3·3-s + 5-s − 8·7-s + 3·9-s + 2·11-s − 13-s − 3·15-s + 3·17-s + 8·19-s + 24·21-s − 23-s + 5·25-s − 9·29-s − 6·33-s − 8·35-s + 4·37-s + 3·39-s + 3·41-s − 43-s + 3·45-s + 11·47-s + 34·49-s − 9·51-s − 3·53-s + 2·55-s − 24·57-s + 9·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 3.02·7-s + 9-s + 0.603·11-s − 0.277·13-s − 0.774·15-s + 0.727·17-s + 1.83·19-s + 5.23·21-s − 0.208·23-s + 25-s − 1.67·29-s − 1.04·33-s − 1.35·35-s + 0.657·37-s + 0.480·39-s + 0.468·41-s − 0.152·43-s + 0.447·45-s + 1.60·47-s + 34/7·49-s − 1.26·51-s − 0.412·53-s + 0.269·55-s − 3.17·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4942577945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4942577945\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.33646702522366867704938640802, −9.853261417665985532128464421874, −9.611362725351919664152496586479, −9.542696485261394383766975109232, −9.012287357229998621241424195668, −8.422539132724706952513108316876, −7.47205495346441206545588922792, −7.24691163535329339986942431308, −6.86755757130513256585438832220, −6.32995388208776792573103976793, −6.05627277259960734746038426820, −5.67137898664213827475036895724, −5.44603287108929398985196088070, −4.84370855560728725893621649019, −3.81664865091574965012795658662, −3.66879317051822943924313715433, −2.98600152902900434212645568168, −2.50904562697765138859608183709, −1.14532152944828503315704370960, −0.44577621937465448763936199651,
0.44577621937465448763936199651, 1.14532152944828503315704370960, 2.50904562697765138859608183709, 2.98600152902900434212645568168, 3.66879317051822943924313715433, 3.81664865091574965012795658662, 4.84370855560728725893621649019, 5.44603287108929398985196088070, 5.67137898664213827475036895724, 6.05627277259960734746038426820, 6.32995388208776792573103976793, 6.86755757130513256585438832220, 7.24691163535329339986942431308, 7.47205495346441206545588922792, 8.422539132724706952513108316876, 9.012287357229998621241424195668, 9.542696485261394383766975109232, 9.611362725351919664152496586479, 9.853261417665985532128464421874, 10.33646702522366867704938640802
