| L(s) = 1 | − 4-s + 2·7-s − 6·13-s − 3·16-s − 19-s + 4·25-s − 2·28-s − 6·31-s + 4·43-s − 2·49-s + 6·52-s + 8·61-s + 7·64-s + 6·67-s + 4·73-s + 76-s + 12·79-s − 12·91-s − 18·97-s − 4·100-s + 24·103-s − 18·109-s − 6·112-s − 14·121-s + 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s − 1.66·13-s − 3/4·16-s − 0.229·19-s + 4/5·25-s − 0.377·28-s − 1.07·31-s + 0.609·43-s − 2/7·49-s + 0.832·52-s + 1.02·61-s + 7/8·64-s + 0.733·67-s + 0.468·73-s + 0.114·76-s + 1.35·79-s − 1.25·91-s − 1.82·97-s − 2/5·100-s + 2.36·103-s − 1.72·109-s − 0.566·112-s − 1.27·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244872874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244872874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.473405990508333938942460162145, −7.979826918997112010936625153086, −7.61815308221120960965862315670, −7.06964187769904693294430669105, −6.80095192435737797106642596261, −6.21210599642288180344551815767, −5.38093432570391435903745350965, −5.20118939724020599935340990965, −4.73935881778659911720376669301, −4.25044865189904014730667776346, −3.74330600536114204681332669925, −2.89600271901229730663785609525, −2.32818830034144713715101440423, −1.76513390485479161742475196655, −0.56983358971280407634685340641,
0.56983358971280407634685340641, 1.76513390485479161742475196655, 2.32818830034144713715101440423, 2.89600271901229730663785609525, 3.74330600536114204681332669925, 4.25044865189904014730667776346, 4.73935881778659911720376669301, 5.20118939724020599935340990965, 5.38093432570391435903745350965, 6.21210599642288180344551815767, 6.80095192435737797106642596261, 7.06964187769904693294430669105, 7.61815308221120960965862315670, 7.979826918997112010936625153086, 8.473405990508333938942460162145
