| L(s) = 1 | − 2-s + 2·3-s + 4-s − 3·5-s − 2·6-s − 8-s − 3·9-s + 3·10-s + 2·12-s + 2·13-s − 6·15-s + 16-s + 3·18-s − 3·20-s − 2·24-s + 4·25-s − 2·26-s − 14·27-s + 6·30-s − 8·31-s − 32-s − 3·36-s − 14·37-s + 4·39-s + 3·40-s − 2·43-s + 9·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 0.353·8-s − 9-s + 0.948·10-s + 0.577·12-s + 0.554·13-s − 1.54·15-s + 1/4·16-s + 0.707·18-s − 0.670·20-s − 0.408·24-s + 4/5·25-s − 0.392·26-s − 2.69·27-s + 1.09·30-s − 1.43·31-s − 0.176·32-s − 1/2·36-s − 2.30·37-s + 0.640·39-s + 0.474·40-s − 0.304·43-s + 1.34·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7565349432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7565349432\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.356510521751481354574700239026, −8.289125051595985253760732717123, −7.70780692916562741973324944754, −7.42270717221216720189715772672, −6.76423132327857768444846951818, −6.43359626777975842229409936906, −5.63829223445030118141243491295, −5.29664428220366180285860647718, −4.63203019707393111980652013047, −3.64028761626013442697591583469, −3.50975251160325053661801805610, −3.26275811523819592310692911169, −2.27866752502195894766164711772, −1.84746323841568047323524015317, −0.45021504622944962260982456300,
0.45021504622944962260982456300, 1.84746323841568047323524015317, 2.27866752502195894766164711772, 3.26275811523819592310692911169, 3.50975251160325053661801805610, 3.64028761626013442697591583469, 4.63203019707393111980652013047, 5.29664428220366180285860647718, 5.63829223445030118141243491295, 6.43359626777975842229409936906, 6.76423132327857768444846951818, 7.42270717221216720189715772672, 7.70780692916562741973324944754, 8.289125051595985253760732717123, 8.356510521751481354574700239026
