| L(s) = 1 | + 3·2-s − 2·3-s + 4·4-s − 6·6-s + 3·8-s + 3·9-s − 8·12-s + 3·16-s + 3·17-s + 9·18-s + 6·19-s + 6·23-s − 6·24-s + 7·25-s − 10·27-s − 3·29-s + 6·32-s + 9·34-s + 12·36-s − 15·37-s + 18·38-s + 9·41-s − 8·43-s + 18·46-s − 6·48-s − 7·49-s + 21·50-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 2·4-s − 2.44·6-s + 1.06·8-s + 9-s − 2.30·12-s + 3/4·16-s + 0.727·17-s + 2.12·18-s + 1.37·19-s + 1.25·23-s − 1.22·24-s + 7/5·25-s − 1.92·27-s − 0.557·29-s + 1.06·32-s + 1.54·34-s + 2·36-s − 2.46·37-s + 2.91·38-s + 1.40·41-s − 1.21·43-s + 2.65·46-s − 0.866·48-s − 49-s + 2.96·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.512848545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.512848545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−13.23256886453846147121212803712, −12.42933955575344049668574404524, −12.21852914455852805368699213239, −11.93860432277066393135330141493, −11.10905189524211363382606267691, −10.93875512283317464077819387751, −10.35129558045231149822368240900, −9.561630726787233406881543566838, −9.233742853214706149257947856107, −8.229845009039480286773078806987, −7.46138868948280234682623686315, −7.07079100798104216271527244937, −6.40872033284167609817910591040, −5.69272654076443095199019384720, −5.39166923775463754527827631491, −4.88674070680156548812257601871, −4.46782341336606192190483743263, −3.33107605046701591035743650285, −3.28836085816980442010612971894, −1.47051711068923491280926552768,
1.47051711068923491280926552768, 3.28836085816980442010612971894, 3.33107605046701591035743650285, 4.46782341336606192190483743263, 4.88674070680156548812257601871, 5.39166923775463754527827631491, 5.69272654076443095199019384720, 6.40872033284167609817910591040, 7.07079100798104216271527244937, 7.46138868948280234682623686315, 8.229845009039480286773078806987, 9.233742853214706149257947856107, 9.561630726787233406881543566838, 10.35129558045231149822368240900, 10.93875512283317464077819387751, 11.10905189524211363382606267691, 11.93860432277066393135330141493, 12.21852914455852805368699213239, 12.42933955575344049668574404524, 13.23256886453846147121212803712
