| L(s) = 1 | − 5-s + 7-s − 7·11-s − 2·13-s − 17-s + 2·19-s − 10·23-s − 5·25-s − 4·31-s − 35-s − 16·37-s + 2·41-s + 43-s + 47-s − 9·49-s + 8·53-s + 7·55-s − 24·59-s + 61-s + 2·65-s − 4·67-s − 5·73-s − 7·77-s − 2·79-s − 8·83-s + 85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 2.11·11-s − 0.554·13-s − 0.242·17-s + 0.458·19-s − 2.08·23-s − 25-s − 0.718·31-s − 0.169·35-s − 2.63·37-s + 0.312·41-s + 0.152·43-s + 0.145·47-s − 9/7·49-s + 1.09·53-s + 0.943·55-s − 3.12·59-s + 0.128·61-s + 0.248·65-s − 0.488·67-s − 0.585·73-s − 0.797·77-s − 0.225·79-s − 0.878·83-s + 0.108·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 56 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.202662513579733759431425021264, −9.149236607541663322133941012394, −8.411515746775814653294218273641, −8.084711086656233871244845227047, −7.66190770990152056405132713219, −7.65332501703342613470881963732, −7.12119993922417367778747557959, −6.50324011812014693129658106225, −5.98630505389494854275615366728, −5.58047950761476600184648104997, −5.05819907741072692268536925856, −4.98739524312174120048643645785, −4.02883090227823545932008734574, −4.00395169995047673255636676435, −3.06940715612023451468291156546, −2.80203519824726207782574024181, −1.94093804883590667586111335914, −1.72180829274827917643095105983, 0, 0,
1.72180829274827917643095105983, 1.94093804883590667586111335914, 2.80203519824726207782574024181, 3.06940715612023451468291156546, 4.00395169995047673255636676435, 4.02883090227823545932008734574, 4.98739524312174120048643645785, 5.05819907741072692268536925856, 5.58047950761476600184648104997, 5.98630505389494854275615366728, 6.50324011812014693129658106225, 7.12119993922417367778747557959, 7.65332501703342613470881963732, 7.66190770990152056405132713219, 8.084711086656233871244845227047, 8.411515746775814653294218273641, 9.149236607541663322133941012394, 9.202662513579733759431425021264
