| L(s) = 1 | + (−0.669 + 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−1.13 + 0.241i)5-s + (−0.309 + 0.951i)6-s + (−1.18 + 2.36i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.579 − 1.00i)10-s + (3.01 − 1.38i)11-s + (−0.5 − 0.866i)12-s + (1.36 + 4.18i)13-s + (−0.961 − 2.46i)14-s + (−0.937 + 0.681i)15-s + (−0.978 + 0.207i)16-s + (0.764 + 0.848i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (0.527 − 0.234i)3-s + (−0.0522 − 0.497i)4-s + (−0.507 + 0.107i)5-s + (−0.126 + 0.388i)6-s + (−0.449 + 0.893i)7-s + (0.286 + 0.207i)8-s + (0.223 − 0.247i)9-s + (0.183 − 0.317i)10-s + (0.908 − 0.417i)11-s + (−0.144 − 0.249i)12-s + (0.377 + 1.16i)13-s + (−0.256 − 0.658i)14-s + (−0.242 + 0.175i)15-s + (−0.244 + 0.0519i)16-s + (0.185 + 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863048 + 0.746531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863048 + 0.746531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 7 | \( 1 + (1.18 - 2.36i)T \) |
| 11 | \( 1 + (-3.01 + 1.38i)T \) |
| good | 5 | \( 1 + (1.13 - 0.241i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 4.18i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.764 - 0.848i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.648 - 6.17i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.0167 - 0.0290i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.81 + 2.04i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.93 - 1.68i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-7.00 - 3.11i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-6.14 - 4.46i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.54T + 43T^{2} \) |
| 47 | \( 1 + (0.417 - 3.97i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (12.0 + 2.55i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (1.13 + 10.7i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-2.77 + 0.589i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (4.65 - 8.05i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.761 - 2.34i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.859 + 8.17i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-5.10 + 5.66i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.60 + 8.01i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.06 + 1.83i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.23 + 13.0i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.45806854781996699536592120514, −9.972598423545106794880294256981, −9.333601068504450684897499987182, −8.435326226716595067737078888043, −7.86156315615651706875015283297, −6.46646505747680670189202538502, −6.14442869691869277026824401419, −4.41630714156510256047551775001, −3.26454549354933187734710166220, −1.64138300139343091470772919849,
0.851685236060884069587495223360, 2.76001089773014150715424932917, 3.78887196780663140097037186248, 4.64275665901945254823607220073, 6.43549375736820026416168202780, 7.44664368927650187915622224689, 8.194239597855453607602429213267, 9.191784486311240454466732851867, 9.931982319517998159092889549514, 10.72431545152944022715480058622
