| L(s) = 1 | − 2·5-s − 4·7-s + 4·11-s − 4·13-s + 4·17-s − 2·19-s + 12·23-s + 3·25-s − 4·29-s − 8·31-s + 8·35-s − 12·37-s + 4·41-s + 4·43-s + 4·47-s + 6·49-s − 4·53-s − 8·55-s − 16·59-s − 16·61-s + 8·65-s − 4·67-s − 16·71-s + 12·73-s − 16·77-s − 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.20·11-s − 1.10·13-s + 0.970·17-s − 0.458·19-s + 2.50·23-s + 3/5·25-s − 0.742·29-s − 1.43·31-s + 1.35·35-s − 1.97·37-s + 0.624·41-s + 0.609·43-s + 0.583·47-s + 6/7·49-s − 0.549·53-s − 1.07·55-s − 2.08·59-s − 2.04·61-s + 0.992·65-s − 0.488·67-s − 1.89·71-s + 1.40·73-s − 1.82·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.291652558942667696356100301950, −8.145276517515521339880094838551, −7.43982365601928693943778971158, −7.23491622359062440134117135010, −6.91281080868657798020790165665, −6.89327091178823281213574633548, −6.02866999723893062138921468795, −5.96464623639124523534730785989, −5.40473088643548048510991073303, −4.84670366568486382460887858958, −4.64643592281404905008676085469, −4.01025103627065891807951145761, −3.54374327199999898693563382993, −3.43390971799688264911219369001, −2.82398481843321878060042684094, −2.59891451645837407559769890165, −1.45552072692072718068603756457, −1.30353614331230833641432697437, 0, 0,
1.30353614331230833641432697437, 1.45552072692072718068603756457, 2.59891451645837407559769890165, 2.82398481843321878060042684094, 3.43390971799688264911219369001, 3.54374327199999898693563382993, 4.01025103627065891807951145761, 4.64643592281404905008676085469, 4.84670366568486382460887858958, 5.40473088643548048510991073303, 5.96464623639124523534730785989, 6.02866999723893062138921468795, 6.89327091178823281213574633548, 6.91281080868657798020790165665, 7.23491622359062440134117135010, 7.43982365601928693943778971158, 8.145276517515521339880094838551, 8.291652558942667696356100301950
