| L(s) = 1 | − 4·5-s + 9-s − 10·11-s − 14·19-s + 5·25-s − 12·31-s − 36·41-s − 4·45-s − 2·49-s + 40·55-s − 8·59-s + 4·61-s + 8·71-s + 28·79-s + 20·89-s + 56·95-s − 10·99-s − 16·101-s − 36·109-s + 47·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s − 3.01·11-s − 3.21·19-s + 25-s − 2.15·31-s − 5.62·41-s − 0.596·45-s − 2/7·49-s + 5.39·55-s − 1.04·59-s + 0.512·61-s + 0.949·71-s + 3.15·79-s + 2.11·89-s + 5.74·95-s − 1.00·99-s − 1.59·101-s − 3.44·109-s + 4.27·121-s + 0.357·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02191133873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02191133873\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 75 T^{2} + 3416 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66812792004240194767648781543, −6.54384092485111405739020370627, −6.53434189670913029978358451985, −5.94769890439093373656069368336, −5.85670671530258892519594174469, −5.35820863686530138571181836972, −5.24999239924552493545243682564, −5.16822638616059567750405277658, −4.82862322259163020898967750542, −4.82084159735300823595300041783, −4.66863527161857467515690422104, −3.96888670374430314587109230220, −3.84958977739073368868564389872, −3.81447691735386871849139050951, −3.78579758902243307153773159391, −3.23041170943549989491939807696, −3.09659087153969776061909428397, −2.67582961653597843891877120540, −2.48153535595883915154938075427, −2.14484910464069273828233967262, −1.82066084015965360069579524556, −1.80674691080792076270632642620, −1.17065167745426718362236006538, −0.24887481811153605779030336358, −0.086552679838786349602083131944,
0.086552679838786349602083131944, 0.24887481811153605779030336358, 1.17065167745426718362236006538, 1.80674691080792076270632642620, 1.82066084015965360069579524556, 2.14484910464069273828233967262, 2.48153535595883915154938075427, 2.67582961653597843891877120540, 3.09659087153969776061909428397, 3.23041170943549989491939807696, 3.78579758902243307153773159391, 3.81447691735386871849139050951, 3.84958977739073368868564389872, 3.96888670374430314587109230220, 4.66863527161857467515690422104, 4.82084159735300823595300041783, 4.82862322259163020898967750542, 5.16822638616059567750405277658, 5.24999239924552493545243682564, 5.35820863686530138571181836972, 5.85670671530258892519594174469, 5.94769890439093373656069368336, 6.53434189670913029978358451985, 6.54384092485111405739020370627, 6.66812792004240194767648781543
