| L(s) = 1 | + 3·3-s − 4·4-s + 4·7-s + 6·9-s − 12·12-s + 2·13-s + 12·16-s + 19-s + 12·21-s − 5·25-s + 9·27-s − 16·28-s − 7·31-s − 24·36-s + 6·39-s + 8·43-s + 36·48-s + 9·49-s − 8·52-s + 3·57-s + 27·61-s + 24·63-s − 32·64-s − 21·67-s − 10·73-s − 15·75-s − 4·76-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2·4-s + 1.51·7-s + 2·9-s − 3.46·12-s + 0.554·13-s + 3·16-s + 0.229·19-s + 2.61·21-s − 25-s + 1.73·27-s − 3.02·28-s − 1.25·31-s − 4·36-s + 0.960·39-s + 1.21·43-s + 5.19·48-s + 9/7·49-s − 1.10·52-s + 0.397·57-s + 3.45·61-s + 3.02·63-s − 4·64-s − 2.56·67-s − 1.17·73-s − 1.73·75-s − 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.089641546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089641546\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−12.11990515335903739985469052413, −11.92589734366127215568968297984, −11.01099641032388266832006557808, −10.64839563086742168272106466912, −9.876686972744191120894742608387, −9.715309858720150862828951638853, −9.087398315469244384431776791870, −8.704746423070926815940792846379, −8.462035230188345410544844106201, −8.012077359490792926095144984368, −7.54439321286363086520148134814, −7.15906306963011642793162970074, −5.75661948492866863806523905206, −5.55287418224715219080595583184, −4.63278912814543631048541880473, −4.31885719901572852730971147120, −3.79521017015359293978764829044, −3.22253338832772130347604381603, −2.09341921989679480560233999901, −1.24027636210832420307063631380,
1.24027636210832420307063631380, 2.09341921989679480560233999901, 3.22253338832772130347604381603, 3.79521017015359293978764829044, 4.31885719901572852730971147120, 4.63278912814543631048541880473, 5.55287418224715219080595583184, 5.75661948492866863806523905206, 7.15906306963011642793162970074, 7.54439321286363086520148134814, 8.012077359490792926095144984368, 8.462035230188345410544844106201, 8.704746423070926815940792846379, 9.087398315469244384431776791870, 9.715309858720150862828951638853, 9.876686972744191120894742608387, 10.64839563086742168272106466912, 11.01099641032388266832006557808, 11.92589734366127215568968297984, 12.11990515335903739985469052413
