| L(s) = 1 | − 4·3-s + 4-s − 4·5-s − 7·7-s + 7·9-s − 2·11-s − 4·12-s − 4·13-s + 16·15-s + 16-s − 17-s − 4·20-s + 28·21-s + 23-s + 6·25-s − 4·27-s − 7·28-s − 4·29-s − 3·31-s + 8·33-s + 28·35-s + 7·36-s + 2·37-s + 16·39-s + 3·41-s − 6·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s − 1.78·5-s − 2.64·7-s + 7/3·9-s − 0.603·11-s − 1.15·12-s − 1.10·13-s + 4.13·15-s + 1/4·16-s − 0.242·17-s − 0.894·20-s + 6.11·21-s + 0.208·23-s + 6/5·25-s − 0.769·27-s − 1.32·28-s − 0.742·29-s − 0.538·31-s + 1.39·33-s + 4.73·35-s + 7/6·36-s + 0.328·37-s + 2.56·39-s + 0.468·41-s − 0.914·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5639 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 172 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 106 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−16.1402896839, −15.7974080705, −15.5615212403, −14.9712481232, −14.3848403799, −13.2307967209, −12.9343416911, −12.5648258390, −12.2454398560, −11.5961102096, −11.4635173947, −11.0086671480, −10.3168556719, −9.93867566050, −9.48460976060, −8.56306739811, −7.70351630672, −7.18799625832, −6.79276578378, −6.27259852296, −5.77648845281, −5.11639590503, −4.35449791581, −3.48217989875, −2.89162655853, 0, 0,
2.89162655853, 3.48217989875, 4.35449791581, 5.11639590503, 5.77648845281, 6.27259852296, 6.79276578378, 7.18799625832, 7.70351630672, 8.56306739811, 9.48460976060, 9.93867566050, 10.3168556719, 11.0086671480, 11.4635173947, 11.5961102096, 12.2454398560, 12.5648258390, 12.9343416911, 13.2307967209, 14.3848403799, 14.9712481232, 15.5615212403, 15.7974080705, 16.1402896839
