| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 3·12-s + 16-s − 6·18-s + 3·24-s − 6·25-s − 9·27-s − 32-s + 6·36-s − 3·48-s − 5·49-s + 6·50-s + 9·54-s + 6·59-s + 64-s − 6·72-s − 22·73-s + 18·75-s + 9·81-s + 3·96-s + 5·98-s − 6·100-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.866·12-s + 1/4·16-s − 1.41·18-s + 0.612·24-s − 6/5·25-s − 1.73·27-s − 0.176·32-s + 36-s − 0.433·48-s − 5/7·49-s + 0.848·50-s + 1.22·54-s + 0.781·59-s + 1/8·64-s − 0.707·72-s − 2.57·73-s + 2.07·75-s + 81-s + 0.306·96-s + 0.505·98-s − 3/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.517100247295835458504747691490, −8.821505062826721493426456901913, −8.346485115427043903647006747524, −7.63319426690666611076647019485, −7.31898637543608553523873125214, −6.71618144641436167847124797791, −6.25301902978888273279670942838, −5.76408672400574303834782049274, −5.35969798694041830995016243882, −4.62439386449710965508478361112, −4.10077034296114052817784037426, −3.22798198529335968856501642727, −2.12578454826062059825187320030, −1.22736777672075120496878718827, 0,
1.22736777672075120496878718827, 2.12578454826062059825187320030, 3.22798198529335968856501642727, 4.10077034296114052817784037426, 4.62439386449710965508478361112, 5.35969798694041830995016243882, 5.76408672400574303834782049274, 6.25301902978888273279670942838, 6.71618144641436167847124797791, 7.31898637543608553523873125214, 7.63319426690666611076647019485, 8.346485115427043903647006747524, 8.821505062826721493426456901913, 9.517100247295835458504747691490
