| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 11-s + 16-s − 6·17-s − 2·19-s − 2·20-s + 22-s − 4·23-s + 2·25-s + 2·31-s + 32-s − 6·34-s + 2·37-s − 2·38-s − 2·40-s + 2·41-s + 4·43-s + 44-s − 4·46-s − 2·47-s − 49-s + 2·50-s − 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s + 2/5·25-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s − 0.316·40-s + 0.312·41-s + 0.609·43-s + 0.150·44-s − 0.589·46-s − 0.291·47-s − 1/7·49-s + 0.282·50-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028259700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028259700\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 170 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−17.8369964058, −17.0434285046, −16.6183702981, −15.9909782834, −15.5390760474, −15.2491223779, −14.6074992840, −14.0382099450, −13.5828602765, −12.8172585107, −12.5505779282, −11.7405240279, −11.4430328108, −10.8364122626, −10.2519107770, −9.35689313101, −8.72782614033, −8.00759119881, −7.43974847057, −6.56734068869, −6.14777980938, −4.97164236309, −4.30225395490, −3.62488793906, −2.33149494969,
2.33149494969, 3.62488793906, 4.30225395490, 4.97164236309, 6.14777980938, 6.56734068869, 7.43974847057, 8.00759119881, 8.72782614033, 9.35689313101, 10.2519107770, 10.8364122626, 11.4430328108, 11.7405240279, 12.5505779282, 12.8172585107, 13.5828602765, 14.0382099450, 14.6074992840, 15.2491223779, 15.5390760474, 15.9909782834, 16.6183702981, 17.0434285046, 17.8369964058
