| L(s) = 1 | − 9·11-s − 4·16-s − 31-s − 9·41-s − 11·61-s − 9·71-s + 101-s + 103-s + 107-s + 109-s + 113-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 36·176-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.71·11-s − 16-s − 0.179·31-s − 1.40·41-s − 1.40·61-s − 1.06·71-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 2.71·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 9 T + 71 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 11 T + 51 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 9 T + 11 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−14.9785059921, −14.6472009970, −13.7687336605, −13.6088707615, −13.2018655633, −12.8346329234, −12.2325581588, −11.8345017056, −11.1186351117, −10.7792940458, −10.4178800696, −9.90624742729, −9.39367718188, −8.69846969028, −8.26915694822, −7.79166438746, −7.27734473627, −6.78303522123, −5.94542404800, −5.45596635627, −4.87771100814, −4.42477254132, −3.30043691941, −2.72257469949, −1.96622350006, 0,
1.96622350006, 2.72257469949, 3.30043691941, 4.42477254132, 4.87771100814, 5.45596635627, 5.94542404800, 6.78303522123, 7.27734473627, 7.79166438746, 8.26915694822, 8.69846969028, 9.39367718188, 9.90624742729, 10.4178800696, 10.7792940458, 11.1186351117, 11.8345017056, 12.2325581588, 12.8346329234, 13.2018655633, 13.6088707615, 13.7687336605, 14.6472009970, 14.9785059921
