| L(s) = 1 | − 4-s + 9-s − 3·16-s + 2·17-s + 4·23-s − 3·25-s − 14·29-s − 36-s + 12·43-s − 6·49-s − 22·53-s + 2·61-s + 7·64-s − 2·68-s − 8·79-s + 81-s − 4·92-s + 3·100-s − 2·101-s − 12·103-s + 12·107-s − 10·113-s + 14·116-s + 2·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/3·9-s − 3/4·16-s + 0.485·17-s + 0.834·23-s − 3/5·25-s − 2.59·29-s − 1/6·36-s + 1.82·43-s − 6/7·49-s − 3.02·53-s + 0.256·61-s + 7/8·64-s − 0.242·68-s − 0.900·79-s + 1/9·81-s − 0.417·92-s + 3/10·100-s − 0.199·101-s − 1.18·103-s + 1.16·107-s − 0.940·113-s + 1.29·116-s + 2/11·121-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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.953496071277202690760628707198, −8.098164585402600429191939375240, −7.76272726756316876468137325109, −7.39615943456456823019184540002, −6.81487182116886192683690490232, −6.30516426105607665657818857518, −5.69690412431117799500913546540, −5.29194749512948700935663905743, −4.68219127611216438190114538860, −4.17481852601793732411806091663, −3.64435551746758327129407971340, −2.99867516247127911393692274227, −2.14073109731314287461912376091, −1.38766286048299990437692235754, 0,
1.38766286048299990437692235754, 2.14073109731314287461912376091, 2.99867516247127911393692274227, 3.64435551746758327129407971340, 4.17481852601793732411806091663, 4.68219127611216438190114538860, 5.29194749512948700935663905743, 5.69690412431117799500913546540, 6.30516426105607665657818857518, 6.81487182116886192683690490232, 7.39615943456456823019184540002, 7.76272726756316876468137325109, 8.098164585402600429191939375240, 8.953496071277202690760628707198
