| L(s) = 1 | − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 2·13-s − 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 25-s − 2·26-s − 28-s − 2·29-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s + 49-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256465797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256465797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.976525779912433683772333851675, −7.39398850413165927730875614743, −6.34518093653275179095390844834, −6.01581061696917710851209529028, −4.92403553365775964167265796866, −4.46792640657225865455095535714, −3.62088779784825026995011416777, −2.28059162307602342837183603744, −1.77611296934943774735876447029, −0.63334663369662260021570161928,
0.63334663369662260021570161928, 1.77611296934943774735876447029, 2.28059162307602342837183603744, 3.62088779784825026995011416777, 4.46792640657225865455095535714, 4.92403553365775964167265796866, 6.01581061696917710851209529028, 6.34518093653275179095390844834, 7.39398850413165927730875614743, 7.976525779912433683772333851675
