| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s + 6·12-s − 4·15-s + 5·16-s − 4·17-s − 6·18-s − 6·20-s − 8·24-s + 3·25-s + 4·27-s − 12·29-s + 8·30-s − 8·31-s − 6·32-s + 8·34-s + 9·36-s + 12·37-s + 8·40-s + 4·41-s + 8·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 0.970·17-s − 1.41·18-s − 1.34·20-s − 1.63·24-s + 3/5·25-s + 0.769·27-s − 2.22·29-s + 1.46·30-s − 1.43·31-s − 1.06·32-s + 1.37·34-s + 3/2·36-s + 1.97·37-s + 1.26·40-s + 0.624·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007642775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007642775\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.468545166225628927029892392593, −8.046518096152990333473559459369, −7.71850915730465861686981178830, −7.42693948834442370517097038804, −7.35197640108061847211878588373, −6.83244680036359649470495983994, −6.54926498629110669761379222494, −5.99352127903112958443237714312, −5.44264134928554155567223675013, −5.40638072026252363427293184397, −4.38658921926225356379316289599, −4.25786046620005344885319922450, −3.74989509798311547860657069625, −3.56463387544722458430995814050, −2.91883604987481559382582709013, −2.38710490289755682511670590389, −2.16089287176042850360167703122, −1.78868874857924587396892691233, −0.71910151700348068788922255322, −0.66933501506302940607173579969,
0.66933501506302940607173579969, 0.71910151700348068788922255322, 1.78868874857924587396892691233, 2.16089287176042850360167703122, 2.38710490289755682511670590389, 2.91883604987481559382582709013, 3.56463387544722458430995814050, 3.74989509798311547860657069625, 4.25786046620005344885319922450, 4.38658921926225356379316289599, 5.40638072026252363427293184397, 5.44264134928554155567223675013, 5.99352127903112958443237714312, 6.54926498629110669761379222494, 6.83244680036359649470495983994, 7.35197640108061847211878588373, 7.42693948834442370517097038804, 7.71850915730465861686981178830, 8.046518096152990333473559459369, 8.468545166225628927029892392593
