| L(s) = 1 | − 3·3-s + 5-s + 4·7-s + 3·9-s + 11-s + 4·13-s − 3·15-s − 3·17-s − 5·19-s − 12·21-s + 3·23-s + 5·25-s − 12·29-s + 31-s − 3·33-s + 4·35-s + 5·37-s − 12·39-s − 20·41-s − 8·43-s + 3·45-s − 47-s + 9·49-s + 9·51-s + 9·53-s + 55-s + 15·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 1.51·7-s + 9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s − 0.727·17-s − 1.14·19-s − 2.61·21-s + 0.625·23-s + 25-s − 2.22·29-s + 0.179·31-s − 0.522·33-s + 0.676·35-s + 0.821·37-s − 1.92·39-s − 3.12·41-s − 1.21·43-s + 0.447·45-s − 0.145·47-s + 9/7·49-s + 1.26·51-s + 1.23·53-s + 0.134·55-s + 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5397859814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5397859814\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−15.45180652684651068427060106111, −14.95036903226213436645933411100, −14.74678045598644006759510080436, −13.64634290129883345826855146101, −13.39493068958989750796753253955, −12.69366067660129257240872095062, −11.86672095023638690750641091131, −11.49841620452065568321808967167, −10.90686188159101635745940938538, −10.89906967327213489504165293918, −10.00406633894279305696986491913, −8.859041642410459046960176387808, −8.613194565707461501191185933750, −7.64440779996722150762856840388, −6.56227563216336332560750120500, −6.31388810281972817628763161017, −5.17063541996666485357012921658, −5.11112800413170815637287925945, −3.87701761008051489127417062745, −1.76846724099755612032672074698,
1.76846724099755612032672074698, 3.87701761008051489127417062745, 5.11112800413170815637287925945, 5.17063541996666485357012921658, 6.31388810281972817628763161017, 6.56227563216336332560750120500, 7.64440779996722150762856840388, 8.613194565707461501191185933750, 8.859041642410459046960176387808, 10.00406633894279305696986491913, 10.89906967327213489504165293918, 10.90686188159101635745940938538, 11.49841620452065568321808967167, 11.86672095023638690750641091131, 12.69366067660129257240872095062, 13.39493068958989750796753253955, 13.64634290129883345826855146101, 14.74678045598644006759510080436, 14.95036903226213436645933411100, 15.45180652684651068427060106111
