L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + (0.5 + 2.59i)7-s − 0.999·8-s + (0.999 + 1.73i)10-s + (2.5 + 4.33i)11-s + 6·13-s + (2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (2 − 3.46i)19-s + 1.99·20-s + 5·22-s + (−2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.188 + 0.981i)7-s − 0.353·8-s + (0.316 + 0.547i)10-s + (0.753 + 1.30i)11-s + 1.66·13-s + (0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.458 − 0.794i)19-s + 0.447·20-s + 1.06·22-s + (−0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53004 + 0.194979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53004 + 0.194979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−11.45688228847364560918921616366, −10.83918973737132585995253991887, −9.488164441171811616303707442350, −8.980879058013000285634068643822, −7.55720783839625913117175217512, −6.58420840546510374209119620213, −5.48322240473819835554297886333, −4.22831925259862427095549030403, −3.16585513149263045912795950809, −1.84019554945691324312473519307,
1.07253666696515717695817810772, 3.77751641580697688276586904034, 4.06368059801025279216366444096, 5.67013401985455615137700811859, 6.39218317429906017235492956273, 7.67629939923188281731416952850, 8.475734715203401914282448175799, 9.071480386247876814458494402307, 10.70261601664224664085062975677, 11.23231644065076563021275437414