| L(s) = 1 | + 2-s − 3-s − 3·5-s − 6-s + 7-s − 8-s − 3·10-s + 5·11-s + 14-s + 3·15-s − 16-s + 4·17-s − 21-s + 5·22-s − 23-s + 24-s + 5·25-s + 27-s − 6·29-s + 3·30-s + 9·31-s − 5·33-s + 4·34-s − 3·35-s + 12·37-s + 3·40-s + 24·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 1.50·11-s + 0.267·14-s + 0.774·15-s − 1/4·16-s + 0.970·17-s − 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s + 25-s + 0.192·27-s − 1.11·29-s + 0.547·30-s + 1.61·31-s − 0.870·33-s + 0.685·34-s − 0.507·35-s + 1.97·37-s + 0.474·40-s + 3.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138176180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138176180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−10.25261162515689573400733939698, −9.650407768669745372666202718997, −9.495037505427235443974712315924, −9.087920251240340788560753482955, −8.357840045885784651778141879406, −8.088741524543364875753841812108, −7.74471412427014847447529700448, −7.24599006001878626005770668970, −6.82450806400615382653206040504, −6.25770126867669343429437759305, −5.72125152001093631461818901784, −5.70981443157699516037835041513, −4.79171287602688970636966379969, −4.30963973791060101563727783530, −4.08377005163099850398828949352, −3.81663452723781749276112096808, −2.91819436233278651467542769673, −2.53069887300620699468188616683, −1.24029064146650868114045456993, −0.75326052828357931026428979232,
0.75326052828357931026428979232, 1.24029064146650868114045456993, 2.53069887300620699468188616683, 2.91819436233278651467542769673, 3.81663452723781749276112096808, 4.08377005163099850398828949352, 4.30963973791060101563727783530, 4.79171287602688970636966379969, 5.70981443157699516037835041513, 5.72125152001093631461818901784, 6.25770126867669343429437759305, 6.82450806400615382653206040504, 7.24599006001878626005770668970, 7.74471412427014847447529700448, 8.088741524543364875753841812108, 8.357840045885784651778141879406, 9.087920251240340788560753482955, 9.495037505427235443974712315924, 9.650407768669745372666202718997, 10.25261162515689573400733939698
