| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 8·11-s − 6·13-s + 6·17-s − 4·21-s − 6·23-s + 6·27-s − 4·29-s − 16·33-s − 6·37-s − 12·39-s + 12·41-s − 6·43-s − 18·47-s + 2·49-s + 12·51-s + 10·53-s − 4·63-s − 18·67-s − 12·69-s − 10·73-s + 16·77-s + 11·81-s − 6·83-s − 8·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 2/3·9-s − 2.41·11-s − 1.66·13-s + 1.45·17-s − 0.872·21-s − 1.25·23-s + 1.15·27-s − 0.742·29-s − 2.78·33-s − 0.986·37-s − 1.92·39-s + 1.87·41-s − 0.914·43-s − 2.62·47-s + 2/7·49-s + 1.68·51-s + 1.37·53-s − 0.503·63-s − 2.19·67-s − 1.44·69-s − 1.17·73-s + 1.82·77-s + 11/9·81-s − 0.658·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9008537337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9008537337\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.766610285977486184781536614746, −9.285152345849520618027460352812, −8.567861705820082026402441543274, −8.557942136292391511888144813265, −7.86472366519306361174980449743, −7.67582681400150810228972534377, −7.44244386212684433193993942438, −7.06202566632976138060279908020, −6.39474630611944680003834418111, −5.80522194710482539098642063165, −5.57794756560665241606885970344, −4.91208335368472627602154805259, −4.79556262901023302769902494080, −4.06173416028383938755914560932, −3.34824530075107269887668813136, −3.11186415975879097431111809649, −2.66418800632813597259367022339, −2.28193055021139060301086659347, −1.62850274420999023814527468902, −0.31641206405672648544789529484,
0.31641206405672648544789529484, 1.62850274420999023814527468902, 2.28193055021139060301086659347, 2.66418800632813597259367022339, 3.11186415975879097431111809649, 3.34824530075107269887668813136, 4.06173416028383938755914560932, 4.79556262901023302769902494080, 4.91208335368472627602154805259, 5.57794756560665241606885970344, 5.80522194710482539098642063165, 6.39474630611944680003834418111, 7.06202566632976138060279908020, 7.44244386212684433193993942438, 7.67582681400150810228972534377, 7.86472366519306361174980449743, 8.557942136292391511888144813265, 8.567861705820082026402441543274, 9.285152345849520618027460352812, 9.766610285977486184781536614746
