| L(s) = 1 | + 2·5-s + 4·7-s − 2·11-s + 3·13-s − 8·17-s + 19-s − 8·23-s + 5·25-s − 4·29-s + 6·31-s + 8·35-s + 37-s − 6·41-s − 11·43-s + 12·47-s + 9·49-s + 12·53-s − 4·55-s + 8·59-s − 12·61-s + 6·65-s + 26·67-s − 20·71-s + 11·73-s − 8·77-s − 6·79-s − 2·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.603·11-s + 0.832·13-s − 1.94·17-s + 0.229·19-s − 1.66·23-s + 25-s − 0.742·29-s + 1.07·31-s + 1.35·35-s + 0.164·37-s − 0.937·41-s − 1.67·43-s + 1.75·47-s + 9/7·49-s + 1.64·53-s − 0.539·55-s + 1.04·59-s − 1.53·61-s + 0.744·65-s + 3.17·67-s − 2.37·71-s + 1.28·73-s − 0.911·77-s − 0.675·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.137520372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137520372\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.981943095355619302911644950567, −8.793208803279708033121274967227, −8.485862272427510297256843424548, −8.088055200662331662843651367220, −7.907974488086105328692353059732, −7.19905446495384035248345791112, −6.74714563035558892810252706902, −6.70603804533987599323116807290, −5.85148063232732749351307855217, −5.74896696538666990325079089328, −5.33653311906833003067491435070, −4.81010935044856428897837040456, −4.39495063421745366945109745250, −4.13966417382138711149414154104, −3.51131856340625239343958472360, −2.80951950098308471228437779420, −2.17818439064586143987430669843, −2.02998938511766737556348125152, −1.46153644938947387268722094698, −0.60301918906794572201867742066,
0.60301918906794572201867742066, 1.46153644938947387268722094698, 2.02998938511766737556348125152, 2.17818439064586143987430669843, 2.80951950098308471228437779420, 3.51131856340625239343958472360, 4.13966417382138711149414154104, 4.39495063421745366945109745250, 4.81010935044856428897837040456, 5.33653311906833003067491435070, 5.74896696538666990325079089328, 5.85148063232732749351307855217, 6.70603804533987599323116807290, 6.74714563035558892810252706902, 7.19905446495384035248345791112, 7.907974488086105328692353059732, 8.088055200662331662843651367220, 8.485862272427510297256843424548, 8.793208803279708033121274967227, 8.981943095355619302911644950567
