| L(s) = 1 | − 3·3-s + 3·4-s + 6·9-s − 9·12-s + 5·16-s − 7·17-s + 2·25-s − 9·27-s − 14·29-s + 18·36-s − 2·43-s − 15·48-s + 4·49-s + 21·51-s + 7·53-s − 2·61-s + 3·64-s − 21·68-s − 6·75-s − 13·79-s + 9·81-s + 42·87-s + 6·100-s − 9·103-s − 27·108-s − 7·113-s − 42·116-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 3/2·4-s + 2·9-s − 2.59·12-s + 5/4·16-s − 1.69·17-s + 2/5·25-s − 1.73·27-s − 2.59·29-s + 3·36-s − 0.304·43-s − 2.16·48-s + 4/7·49-s + 2.94·51-s + 0.961·53-s − 0.256·61-s + 3/8·64-s − 2.54·68-s − 0.692·75-s − 1.46·79-s + 81-s + 4.50·87-s + 3/5·100-s − 0.886·103-s − 2.59·108-s − 0.658·113-s − 3.89·116-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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 183 T^{2} + p^{2} T^{4} \) |
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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
−8.787863236883086406149932520000, −8.036773779140664832559129548663, −7.38243935469520342027402619483, −7.13770254184426291478851279212, −6.77617205003545719065967830745, −6.28126254394220568602641951968, −5.88707827937703953258524406489, −5.44389867651834545312641371627, −4.91718345396284469961087851006, −4.20493081115552846681698548394, −3.73475539424976550735743662389, −2.70291190818456450517995365865, −2.06660346250998678388863867130, −1.39261442901813238085150012005, 0,
1.39261442901813238085150012005, 2.06660346250998678388863867130, 2.70291190818456450517995365865, 3.73475539424976550735743662389, 4.20493081115552846681698548394, 4.91718345396284469961087851006, 5.44389867651834545312641371627, 5.88707827937703953258524406489, 6.28126254394220568602641951968, 6.77617205003545719065967830745, 7.13770254184426291478851279212, 7.38243935469520342027402619483, 8.036773779140664832559129548663, 8.787863236883086406149932520000
