| L(s) = 1 | + 3-s − 2·4-s − 3·7-s − 6·11-s − 2·12-s + 7·13-s + 6·19-s − 3·21-s + 6·23-s − 2·25-s − 27-s + 6·28-s − 6·29-s − 6·33-s + 7·39-s − 12·41-s + 43-s + 12·44-s − 49-s − 14·52-s + 24·53-s + 6·57-s − 6·59-s − 61-s + 8·64-s + 15·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.13·7-s − 1.80·11-s − 0.577·12-s + 1.94·13-s + 1.37·19-s − 0.654·21-s + 1.25·23-s − 2/5·25-s − 0.192·27-s + 1.13·28-s − 1.11·29-s − 1.04·33-s + 1.12·39-s − 1.87·41-s + 0.152·43-s + 1.80·44-s − 1/7·49-s − 1.94·52-s + 3.29·53-s + 0.794·57-s − 0.781·59-s − 0.128·61-s + 64-s + 1.83·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5420096419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5420096419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.40034926286238632389372103235, −15.88808616152548985528097002510, −15.55977393109343258918665426919, −14.91846109816130859091273293097, −13.96419287562541034555919049480, −13.47313273791666928233950183414, −13.11314330705065684621775478908, −13.04619436231722837529409926035, −11.78493119263088006316411982323, −11.10551651873457236367808656979, −10.26712380940950273475966635755, −9.818472480147079412536725966363, −8.848690190067500122946250492219, −8.739124385087970338720574082298, −7.75582342999115831642782779487, −6.96894414488669299222772554564, −5.79122345317736035221054158129, −5.16881605044338984679923592521, −3.79028449147413468164063210595, −3.03451756138043372419512126667,
3.03451756138043372419512126667, 3.79028449147413468164063210595, 5.16881605044338984679923592521, 5.79122345317736035221054158129, 6.96894414488669299222772554564, 7.75582342999115831642782779487, 8.739124385087970338720574082298, 8.848690190067500122946250492219, 9.818472480147079412536725966363, 10.26712380940950273475966635755, 11.10551651873457236367808656979, 11.78493119263088006316411982323, 13.04619436231722837529409926035, 13.11314330705065684621775478908, 13.47313273791666928233950183414, 13.96419287562541034555919049480, 14.91846109816130859091273293097, 15.55977393109343258918665426919, 15.88808616152548985528097002510, 16.40034926286238632389372103235
