| L(s) = 1 | + 3·5-s − 7-s − 3·11-s − 2·13-s + 3·17-s + 19-s + 3·23-s + 5·25-s − 6·29-s − 14·31-s − 3·35-s + 37-s + 6·41-s + 4·43-s + 18·47-s − 6·49-s + 3·53-s − 9·55-s − 18·59-s − 2·61-s − 6·65-s − 14·67-s + 73-s + 3·77-s − 26·79-s + 12·83-s + 9·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 25-s − 1.11·29-s − 2.51·31-s − 0.507·35-s + 0.164·37-s + 0.937·41-s + 0.609·43-s + 2.62·47-s − 6/7·49-s + 0.412·53-s − 1.21·55-s − 2.34·59-s − 0.256·61-s − 0.744·65-s − 1.71·67-s + 0.117·73-s + 0.341·77-s − 2.92·79-s + 1.31·83-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113067823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113067823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.454421817389738885184196098628, −8.933723875387361208895891168707, −8.749632798973105898664159318092, −7.84418075478690581906237320106, −7.52354974418240833182804897333, −7.33482152502046658107579144206, −7.16146112221052080408843199123, −6.21582240130651130170105828829, −6.01002884622143742287335063475, −5.74898014877545787872330026735, −5.40783223089288648942173295959, −4.86983705324186474827573424052, −4.59204272889981145146605285164, −3.80202708078784072634909264107, −3.43536765771688009791699329261, −2.83268423317282580379114461510, −2.49097837653414290580284889324, −1.89956304994154289876923379514, −1.44135103368453333338360190837, −0.48748414163375360707611253225,
0.48748414163375360707611253225, 1.44135103368453333338360190837, 1.89956304994154289876923379514, 2.49097837653414290580284889324, 2.83268423317282580379114461510, 3.43536765771688009791699329261, 3.80202708078784072634909264107, 4.59204272889981145146605285164, 4.86983705324186474827573424052, 5.40783223089288648942173295959, 5.74898014877545787872330026735, 6.01002884622143742287335063475, 6.21582240130651130170105828829, 7.16146112221052080408843199123, 7.33482152502046658107579144206, 7.52354974418240833182804897333, 7.84418075478690581906237320106, 8.749632798973105898664159318092, 8.933723875387361208895891168707, 9.454421817389738885184196098628
