| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 3·9-s − 2·12-s + 16-s + 2·17-s + 3·18-s − 2·24-s − 6·25-s − 4·27-s + 32-s + 2·34-s + 3·36-s − 12·41-s + 8·43-s − 2·48-s + 10·49-s − 6·50-s − 4·51-s − 4·54-s + 16·59-s + 64-s + 2·68-s + 3·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 9-s − 0.577·12-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.408·24-s − 6/5·25-s − 0.769·27-s + 0.176·32-s + 0.342·34-s + 1/2·36-s − 1.87·41-s + 1.21·43-s − 0.288·48-s + 10/7·49-s − 0.848·50-s − 0.560·51-s − 0.544·54-s + 2.08·59-s + 1/8·64-s + 0.242·68-s + 0.353·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800787128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800787128\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.661140335457030953268886751361, −8.234369008581860367630742160110, −7.65049983315333815787742157938, −7.23007339071516806375060687388, −6.78961807332716027516059567573, −6.31545545210484520681532449235, −5.77190626255755447945500332687, −5.46817165791435838365201502987, −5.02683125206524074860344164800, −4.39174104486575572646924771442, −3.87028361929848019243608003645, −3.41714102710223844709096907143, −2.45289756185632805884410446281, −1.79327144070165485772745385168, −0.74719969981905527974315558451,
0.74719969981905527974315558451, 1.79327144070165485772745385168, 2.45289756185632805884410446281, 3.41714102710223844709096907143, 3.87028361929848019243608003645, 4.39174104486575572646924771442, 5.02683125206524074860344164800, 5.46817165791435838365201502987, 5.77190626255755447945500332687, 6.31545545210484520681532449235, 6.78961807332716027516059567573, 7.23007339071516806375060687388, 7.65049983315333815787742157938, 8.234369008581860367630742160110, 8.661140335457030953268886751361
