| L(s) = 1 | − 2·2-s + 2·4-s − 10·7-s + 9-s + 20·14-s − 4·16-s − 2·17-s − 2·18-s − 8·23-s − 25-s − 20·28-s − 12·31-s + 8·32-s + 4·34-s + 2·36-s + 16·46-s − 18·47-s + 61·49-s + 2·50-s + 24·62-s − 10·63-s − 8·64-s − 4·68-s − 24·71-s − 22·73-s + 32·79-s + 81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 3.77·7-s + 1/3·9-s + 5.34·14-s − 16-s − 0.485·17-s − 0.471·18-s − 1.66·23-s − 1/5·25-s − 3.77·28-s − 2.15·31-s + 1.41·32-s + 0.685·34-s + 1/3·36-s + 2.35·46-s − 2.62·47-s + 61/7·49-s + 0.282·50-s + 3.04·62-s − 1.25·63-s − 64-s − 0.485·68-s − 2.84·71-s − 2.57·73-s + 3.60·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.836427381447004052087018681826, −8.271040340893799936478523446385, −7.47848778675296589128891353641, −7.28520568094672906875359765978, −6.63357575179929515782660247307, −6.41173993136661837405396202601, −6.00986516238713156993238557131, −5.34859318467803908390516046374, −4.18915210248794032182293713544, −3.81591641511052084615077615666, −3.19105931349205486017246714240, −2.62642387931761158548684226837, −1.69858004562926179744598539363, 0, 0,
1.69858004562926179744598539363, 2.62642387931761158548684226837, 3.19105931349205486017246714240, 3.81591641511052084615077615666, 4.18915210248794032182293713544, 5.34859318467803908390516046374, 6.00986516238713156993238557131, 6.41173993136661837405396202601, 6.63357575179929515782660247307, 7.28520568094672906875359765978, 7.47848778675296589128891353641, 8.271040340893799936478523446385, 8.836427381447004052087018681826
