| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 6·5-s + 6·6-s − 3·8-s + 3·9-s + 18·10-s − 8·12-s + 13-s + 12·15-s + 3·16-s − 9·18-s − 24·20-s − 12·23-s + 6·24-s + 17·25-s − 3·26-s − 10·27-s − 6·29-s − 36·30-s − 6·32-s + 12·36-s − 7·37-s − 2·39-s + 18·40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 2.68·5-s + 2.44·6-s − 1.06·8-s + 9-s + 5.69·10-s − 2.30·12-s + 0.277·13-s + 3.09·15-s + 3/4·16-s − 2.12·18-s − 5.36·20-s − 2.50·23-s + 1.22·24-s + 17/5·25-s − 0.588·26-s − 1.92·27-s − 1.11·29-s − 6.57·30-s − 1.06·32-s + 2·36-s − 1.15·37-s − 0.320·39-s + 2.84·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 181 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 155 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−12.06341419935979016490629186866, −11.96135058626500565907493950917, −11.18756730611411518055285431236, −11.05682417095486864256001019689, −10.29739449963896145137292694021, −10.03582692836626050470892461266, −9.395283165186731653838726107372, −8.727485969522545272599660312020, −8.250222609339061762491273595876, −7.85636988939975860551993669273, −7.57949174350019882839721130445, −7.06175644560921893438675027020, −6.28573311260063556660968840581, −5.58472381541945519530742661767, −4.67448396968972048213984353027, −3.75003937096237772145054588655, −3.71092782667346643797561436493, −1.66043516365082145015958815276, 0, 0,
1.66043516365082145015958815276, 3.71092782667346643797561436493, 3.75003937096237772145054588655, 4.67448396968972048213984353027, 5.58472381541945519530742661767, 6.28573311260063556660968840581, 7.06175644560921893438675027020, 7.57949174350019882839721130445, 7.85636988939975860551993669273, 8.250222609339061762491273595876, 8.727485969522545272599660312020, 9.395283165186731653838726107372, 10.03582692836626050470892461266, 10.29739449963896145137292694021, 11.05682417095486864256001019689, 11.18756730611411518055285431236, 11.96135058626500565907493950917, 12.06341419935979016490629186866
