| L(s) = 1 | − 2-s − 3·5-s − 3·7-s + 8-s + 3·10-s + 3·14-s − 16-s − 6·17-s − 8·19-s + 4·25-s + 3·29-s + 31-s + 6·34-s + 9·35-s − 9·37-s + 8·38-s − 3·40-s + 3·41-s + 3·43-s + 2·49-s − 4·50-s − 6·53-s − 3·56-s − 3·58-s + 6·59-s + 4·61-s − 62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 1.13·7-s + 0.353·8-s + 0.948·10-s + 0.801·14-s − 1/4·16-s − 1.45·17-s − 1.83·19-s + 4/5·25-s + 0.557·29-s + 0.179·31-s + 1.02·34-s + 1.52·35-s − 1.47·37-s + 1.29·38-s − 0.474·40-s + 0.468·41-s + 0.457·43-s + 2/7·49-s − 0.565·50-s − 0.824·53-s − 0.400·56-s − 0.393·58-s + 0.781·59-s + 0.512·61-s − 0.127·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 61 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−17.1672541538, −16.5852079085, −16.0075467398, −15.7479579316, −15.3763842293, −14.7758520448, −14.1724228380, −13.4318073958, −12.9545554092, −12.5685939504, −11.9844812033, −11.3458856759, −10.7825797908, −10.4243102586, −9.69264333653, −8.98605946822, −8.63074685500, −8.08524264284, −7.36861248226, −6.59460473929, −6.40012697449, −5.02925970663, −4.20323236584, −3.66402428813, −2.42459333467, 0,
2.42459333467, 3.66402428813, 4.20323236584, 5.02925970663, 6.40012697449, 6.59460473929, 7.36861248226, 8.08524264284, 8.63074685500, 8.98605946822, 9.69264333653, 10.4243102586, 10.7825797908, 11.3458856759, 11.9844812033, 12.5685939504, 12.9545554092, 13.4318073958, 14.1724228380, 14.7758520448, 15.3763842293, 15.7479579316, 16.0075467398, 16.5852079085, 17.1672541538
