| L(s) = 1 | + 2·5-s + 4·7-s − 4·11-s − 4·13-s − 8·17-s − 4·19-s − 4·23-s + 3·25-s − 10·29-s − 4·31-s + 8·35-s − 6·41-s − 16·43-s − 4·47-s + 49-s − 12·53-s − 8·55-s − 8·59-s − 10·61-s − 8·65-s + 16·67-s − 12·71-s − 16·77-s + 24·79-s + 16·83-s − 16·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.20·11-s − 1.10·13-s − 1.94·17-s − 0.917·19-s − 0.834·23-s + 3/5·25-s − 1.85·29-s − 0.718·31-s + 1.35·35-s − 0.937·41-s − 2.43·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s − 1.07·55-s − 1.04·59-s − 1.28·61-s − 0.992·65-s + 1.95·67-s − 1.42·71-s − 1.82·77-s + 2.70·79-s + 1.75·83-s − 1.73·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 195 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 203 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.406429303635452348647648700377, −8.145693696437440781237211565717, −7.64312649560779704318168366924, −7.60534244089851994294518701108, −6.86777526675223856192149412787, −6.66183823242826621240456410309, −6.15204437947550932623140120046, −5.88507817170482106462117556534, −5.09107219963562564572480395396, −5.09053823878496944611103991354, −4.67482918049466163388921891146, −4.56204830371029114281456510320, −3.52189789079067649082551793595, −3.47907348552524546368203961946, −2.45119853347985243936707860729, −2.22849986527587114548761551231, −1.80519102729250395352742362289, −1.62957260170116153096483168890, 0, 0,
1.62957260170116153096483168890, 1.80519102729250395352742362289, 2.22849986527587114548761551231, 2.45119853347985243936707860729, 3.47907348552524546368203961946, 3.52189789079067649082551793595, 4.56204830371029114281456510320, 4.67482918049466163388921891146, 5.09053823878496944611103991354, 5.09107219963562564572480395396, 5.88507817170482106462117556534, 6.15204437947550932623140120046, 6.66183823242826621240456410309, 6.86777526675223856192149412787, 7.60534244089851994294518701108, 7.64312649560779704318168366924, 8.145693696437440781237211565717, 8.406429303635452348647648700377
