| L(s) = 1 | + 3·4-s − 3·9-s + 5·16-s − 8·17-s − 12·23-s − 2·25-s + 4·29-s − 9·36-s − 4·43-s + 2·49-s − 8·53-s − 20·61-s + 3·64-s − 24·68-s − 4·79-s + 9·81-s − 36·92-s − 6·100-s − 12·101-s + 12·103-s + 24·107-s − 4·113-s + 12·116-s − 10·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 9-s + 5/4·16-s − 1.94·17-s − 2.50·23-s − 2/5·25-s + 0.742·29-s − 3/2·36-s − 0.609·43-s + 2/7·49-s − 1.09·53-s − 2.56·61-s + 3/8·64-s − 2.91·68-s − 0.450·79-s + 81-s − 3.75·92-s − 3/5·100-s − 1.19·101-s + 1.18·103-s + 2.32·107-s − 0.376·113-s + 1.11·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
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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
−8.575844740748349945305426282938, −8.198281045973500546936388703785, −7.72134426742478134618804855927, −7.32118433611174634150918371299, −6.54197080240916500609896620000, −6.38759731341826738900181183271, −6.01903020325965004900280082875, −5.47658604109879293695853107016, −4.61451168043936619757087355228, −4.21902181999452119321192089413, −3.35765943021230287908373973332, −2.81142629228725734166133721573, −2.13774635553191506956791143664, −1.78364366196035525018795883062, 0,
1.78364366196035525018795883062, 2.13774635553191506956791143664, 2.81142629228725734166133721573, 3.35765943021230287908373973332, 4.21902181999452119321192089413, 4.61451168043936619757087355228, 5.47658604109879293695853107016, 6.01903020325965004900280082875, 6.38759731341826738900181183271, 6.54197080240916500609896620000, 7.32118433611174634150918371299, 7.72134426742478134618804855927, 8.198281045973500546936388703785, 8.575844740748349945305426282938
