| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5·9-s + 14-s + 16-s + 5·18-s + 9·23-s − 25-s + 28-s − 6·31-s + 32-s + 5·36-s − 3·41-s + 9·46-s − 6·49-s − 50-s + 56-s − 6·62-s + 5·63-s + 64-s + 5·72-s − 12·73-s + 20·79-s + 16·81-s − 3·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 5/3·9-s + 0.267·14-s + 1/4·16-s + 1.17·18-s + 1.87·23-s − 1/5·25-s + 0.188·28-s − 1.07·31-s + 0.176·32-s + 5/6·36-s − 0.468·41-s + 1.32·46-s − 6/7·49-s − 0.141·50-s + 0.133·56-s − 0.762·62-s + 0.629·63-s + 1/8·64-s + 0.589·72-s − 1.40·73-s + 2.25·79-s + 16/9·81-s − 0.331·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.178889923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.178889923\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.258916082147760122973227186270, −8.855987333701881029367042847493, −8.182076831691115753107010144026, −7.60521586312734445037125278111, −7.29065931940753992156447321196, −6.72530231073670518644013583212, −6.50276811898959881094345167923, −5.50158182534795143028508200978, −5.23502964567619810803529997303, −4.57008330055734581911530143534, −4.18660809516532812552822809888, −3.51351867181272689731806603363, −2.85873554915655646962545333048, −1.87927809084846638710062448750, −1.25273099102646431642101040776,
1.25273099102646431642101040776, 1.87927809084846638710062448750, 2.85873554915655646962545333048, 3.51351867181272689731806603363, 4.18660809516532812552822809888, 4.57008330055734581911530143534, 5.23502964567619810803529997303, 5.50158182534795143028508200978, 6.50276811898959881094345167923, 6.72530231073670518644013583212, 7.29065931940753992156447321196, 7.60521586312734445037125278111, 8.182076831691115753107010144026, 8.855987333701881029367042847493, 9.258916082147760122973227186270
