| L(s) = 1 | − 2·3-s − 4·5-s + 2·9-s − 2·11-s + 4·13-s + 8·15-s + 8·17-s − 10·19-s + 8·25-s − 6·27-s − 12·29-s − 16·31-s + 4·33-s + 4·37-s − 8·39-s − 6·43-s − 8·45-s − 2·49-s − 16·51-s − 4·53-s + 8·55-s + 20·57-s + 6·59-s + 12·61-s − 16·65-s − 6·67-s − 16·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 2/3·9-s − 0.603·11-s + 1.10·13-s + 2.06·15-s + 1.94·17-s − 2.29·19-s + 8/5·25-s − 1.15·27-s − 2.22·29-s − 2.87·31-s + 0.696·33-s + 0.657·37-s − 1.28·39-s − 0.914·43-s − 1.19·45-s − 2/7·49-s − 2.24·51-s − 0.549·53-s + 1.07·55-s + 2.64·57-s + 0.781·59-s + 1.53·61-s − 1.98·65-s − 0.733·67-s − 1.84·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−9.689360619551227449509741235991, −9.447651134227729078341170471500, −8.755096790324712371861479237440, −8.435780683320674384609438935994, −7.88737052994389186279896323492, −7.77891682488063234786745697551, −7.16217914091863556617504823248, −6.96779062635909559457398390557, −6.23097237621964385416743819899, −5.72812836812556639272719159386, −5.52106267555180265475607362931, −5.11623368886745793290367160927, −4.19942307312353317363870580401, −3.94921456251181557653383699816, −3.70290815934971434515936580770, −3.09047139588442463285711751689, −2.02325812911198847053479850855, −1.37179631026388951266522490264, 0, 0,
1.37179631026388951266522490264, 2.02325812911198847053479850855, 3.09047139588442463285711751689, 3.70290815934971434515936580770, 3.94921456251181557653383699816, 4.19942307312353317363870580401, 5.11623368886745793290367160927, 5.52106267555180265475607362931, 5.72812836812556639272719159386, 6.23097237621964385416743819899, 6.96779062635909559457398390557, 7.16217914091863556617504823248, 7.77891682488063234786745697551, 7.88737052994389186279896323492, 8.435780683320674384609438935994, 8.755096790324712371861479237440, 9.447651134227729078341170471500, 9.689360619551227449509741235991
