| L(s) = 1 | − 10·7-s − 3·9-s + 4·16-s + 14·19-s − 14·31-s − 2·37-s − 6·43-s + 59·49-s + 30·63-s + 32·67-s − 34·73-s − 38·97-s + 38·109-s − 40·112-s + 127-s + 131-s − 140·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 42·171-s + ⋯ |
| L(s) = 1 | − 3.77·7-s − 9-s + 16-s + 3.21·19-s − 2.51·31-s − 0.328·37-s − 0.914·43-s + 59/7·49-s + 3.77·63-s + 3.90·67-s − 3.97·73-s − 3.85·97-s + 3.63·109-s − 3.77·112-s + 0.0887·127-s + 0.0873·131-s − 12.1·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 3.21·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2814745716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2814745716\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 19 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−12.60526945583769593725335756279, −12.04244590137973943733356661932, −11.68233344115948147884910941145, −11.27827400375444776803415132348, −11.20671029059133093468554275184, −10.33799096166553589972307571081, −10.28492346517608797924035033166, −9.878455700632552780346137377328, −9.662357628454430853550576199998, −9.311099632916972730542005383458, −9.266331349291400128893358857922, −8.614704473648495175511787933697, −8.407971758694125939444127237067, −7.49888675486433736155097520690, −7.45618645320222714468884015368, −6.87588616744967647746205702468, −6.84682932914977747208997913860, −6.04231259098432226043157725372, −5.83187567293754489209147142198, −5.50669007436295339404409935203, −5.18384665668504267031376994747, −3.82871138839055860330868972501, −3.59850093997033233524973792826, −3.07843900728166971573586321345, −2.90998648777473488123802691863,
2.90998648777473488123802691863, 3.07843900728166971573586321345, 3.59850093997033233524973792826, 3.82871138839055860330868972501, 5.18384665668504267031376994747, 5.50669007436295339404409935203, 5.83187567293754489209147142198, 6.04231259098432226043157725372, 6.84682932914977747208997913860, 6.87588616744967647746205702468, 7.45618645320222714468884015368, 7.49888675486433736155097520690, 8.407971758694125939444127237067, 8.614704473648495175511787933697, 9.266331349291400128893358857922, 9.311099632916972730542005383458, 9.662357628454430853550576199998, 9.878455700632552780346137377328, 10.28492346517608797924035033166, 10.33799096166553589972307571081, 11.20671029059133093468554275184, 11.27827400375444776803415132348, 11.68233344115948147884910941145, 12.04244590137973943733356661932, 12.60526945583769593725335756279
