| L(s) = 1 | − 2·4-s − 12·11-s + 4·16-s + 12·17-s − 2·19-s + 10·25-s + 20·43-s + 24·44-s + 14·49-s − 8·64-s − 24·68-s − 4·73-s + 4·76-s + 36·83-s − 20·100-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s − 40·172-s + ⋯ |
| L(s) = 1 | − 4-s − 3.61·11-s + 16-s + 2.91·17-s − 0.458·19-s + 2·25-s + 3.04·43-s + 3.61·44-s + 2·49-s − 64-s − 2.91·68-s − 0.468·73-s + 0.458·76-s + 3.95·83-s − 2·100-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s − 3.04·172-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
| \(L(1)\) |
\(\approx\) |
\(1.372391809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372391809\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.887743557259295357306403674278, −9.427356820178783717888971823000, −8.842523639411931260728442236845, −8.680592105246530502545248350416, −8.012294152964341360854348563966, −7.77478870733858441459102325797, −7.49784233994456264546582566456, −7.39146760486204370743379766954, −6.33556632328015156460050011174, −5.89214249422208076149110800219, −5.34176530067946598830117165589, −5.29240550147761721819958815602, −4.97498630802277822528476478255, −4.34304867614758680394244821808, −3.67849533585647154438575924998, −3.20301448067776836139419303212, −2.58337066785736153446606470666, −2.48981731787086850507128839995, −1.07669308605252480504804906661, −0.58964057398262942127452448794,
0.58964057398262942127452448794, 1.07669308605252480504804906661, 2.48981731787086850507128839995, 2.58337066785736153446606470666, 3.20301448067776836139419303212, 3.67849533585647154438575924998, 4.34304867614758680394244821808, 4.97498630802277822528476478255, 5.29240550147761721819958815602, 5.34176530067946598830117165589, 5.89214249422208076149110800219, 6.33556632328015156460050011174, 7.39146760486204370743379766954, 7.49784233994456264546582566456, 7.77478870733858441459102325797, 8.012294152964341360854348563966, 8.680592105246530502545248350416, 8.842523639411931260728442236845, 9.427356820178783717888971823000, 9.887743557259295357306403674278
