L(s) = 1 | + 2-s + 4-s − 4·5-s + 2·7-s + 8-s − 4·10-s + 2·14-s + 16-s − 17-s − 4·20-s + 6·23-s + 11·25-s + 2·28-s − 2·29-s + 4·31-s + 32-s − 34-s − 8·35-s + 2·37-s − 4·40-s − 6·41-s + 4·43-s + 6·46-s + 6·47-s − 3·49-s + 11·50-s − 8·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s + 0.353·8-s − 1.26·10-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.894·20-s + 1.25·23-s + 11/5·25-s + 0.377·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.35·35-s + 0.328·37-s − 0.632·40-s − 0.937·41-s + 0.609·43-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 1.55·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.665430579\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.665430579\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.96187105040578, −14.44269917098991, −13.98455567158238, −13.10389050562664, −12.88746128574261, −12.08093768581224, −11.81335303970030, −11.32557447465787, −10.86342420682284, −10.47654614850827, −9.490119654054022, −8.794582729203753, −8.345034709714413, −7.685623577919244, −7.435226316041863, −6.763687508631081, −6.158816566220627, −5.230844817607960, −4.717671939953681, −4.397701907202875, −3.613143329599662, −3.185082721491055, −2.400048565261870, −1.392467857726960, −0.5664085747496719,
0.5664085747496719, 1.392467857726960, 2.400048565261870, 3.185082721491055, 3.613143329599662, 4.397701907202875, 4.717671939953681, 5.230844817607960, 6.158816566220627, 6.763687508631081, 7.435226316041863, 7.685623577919244, 8.345034709714413, 8.794582729203753, 9.490119654054022, 10.47654614850827, 10.86342420682284, 11.32557447465787, 11.81335303970030, 12.08093768581224, 12.88746128574261, 13.10389050562664, 13.98455567158238, 14.44269917098991, 14.96187105040578