| L(s) = 1 | − 3·3-s + 2·5-s − 2·7-s + 6·9-s − 5·11-s − 4·13-s − 6·15-s + 2·17-s − 10·19-s + 6·21-s − 4·23-s + 5·25-s − 9·27-s + 6·29-s + 15·33-s − 4·35-s − 20·37-s + 12·39-s + 3·41-s + 9·43-s + 12·45-s + 8·47-s + 7·49-s − 6·51-s − 24·53-s − 10·55-s + 30·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 0.755·7-s + 2·9-s − 1.50·11-s − 1.10·13-s − 1.54·15-s + 0.485·17-s − 2.29·19-s + 1.30·21-s − 0.834·23-s + 25-s − 1.73·27-s + 1.11·29-s + 2.61·33-s − 0.676·35-s − 3.28·37-s + 1.92·39-s + 0.468·41-s + 1.37·43-s + 1.78·45-s + 1.16·47-s + 49-s − 0.840·51-s − 3.29·53-s − 1.34·55-s + 3.97·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.804291483074349521783022380196, −9.318552473057008196926696524164, −8.823335533670322810874373334040, −8.475289211636257676003047722987, −7.63201633977574675939564412610, −7.58227501404070926618749906799, −6.83396018525972418377085582285, −6.62078756146929241508908590457, −6.10587650465506011834093408915, −5.87049564235436623684334630471, −5.33737278878949744876775184431, −5.06154608558179688735520907327, −4.42072334479193272215695822855, −4.26995340203270568033917507602, −3.19008242669462900906058746056, −2.69535198938225250825949483938, −2.09751376058906691106713815825, −1.42297213921239655469885219623, 0, 0,
1.42297213921239655469885219623, 2.09751376058906691106713815825, 2.69535198938225250825949483938, 3.19008242669462900906058746056, 4.26995340203270568033917507602, 4.42072334479193272215695822855, 5.06154608558179688735520907327, 5.33737278878949744876775184431, 5.87049564235436623684334630471, 6.10587650465506011834093408915, 6.62078756146929241508908590457, 6.83396018525972418377085582285, 7.58227501404070926618749906799, 7.63201633977574675939564412610, 8.475289211636257676003047722987, 8.823335533670322810874373334040, 9.318552473057008196926696524164, 9.804291483074349521783022380196
