| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s + 16-s + 2·18-s + 8·23-s − 2·25-s + 8·29-s − 32-s − 2·36-s − 16·37-s − 8·46-s − 7·49-s + 2·50-s + 16·53-s − 8·58-s + 64-s − 24·71-s + 2·72-s + 16·74-s − 32·79-s − 5·81-s + 8·92-s + 7·98-s − 2·100-s − 16·106-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 1/4·16-s + 0.471·18-s + 1.66·23-s − 2/5·25-s + 1.48·29-s − 0.176·32-s − 1/3·36-s − 2.63·37-s − 1.17·46-s − 49-s + 0.282·50-s + 2.19·53-s − 1.05·58-s + 1/8·64-s − 2.84·71-s + 0.235·72-s + 1.85·74-s − 3.60·79-s − 5/9·81-s + 0.834·92-s + 0.707·98-s − 1/5·100-s − 1.55·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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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
−8.312643462144221960724403117674, −7.39882727372647374248012947861, −7.28815113266216039232851219178, −6.91871579106931057797033104353, −6.27026212550748067984186009030, −5.88045142773096109817184061456, −5.35948295955480095458297786653, −4.89534372790661416769740588179, −4.36423285347045925412696055403, −3.59825789028225141662724729996, −3.01851490629752752797026746586, −2.71374431693977615462851425493, −1.80671257071984273314943130579, −1.12565693679380310126026867196, 0,
1.12565693679380310126026867196, 1.80671257071984273314943130579, 2.71374431693977615462851425493, 3.01851490629752752797026746586, 3.59825789028225141662724729996, 4.36423285347045925412696055403, 4.89534372790661416769740588179, 5.35948295955480095458297786653, 5.88045142773096109817184061456, 6.27026212550748067984186009030, 6.91871579106931057797033104353, 7.28815113266216039232851219178, 7.39882727372647374248012947861, 8.312643462144221960724403117674
