L(s) = 1 | + (1.38 + 0.264i)2-s − 3-s + (1.85 + 0.735i)4-s + (−1.38 − 0.264i)6-s + 0.941i·7-s + (2.38 + 1.51i)8-s + 9-s − 4.49i·11-s + (−1.85 − 0.735i)12-s + 5.55·13-s + (−0.249 + 1.30i)14-s + (2.91 + 2.73i)16-s + 7.55i·17-s + (1.38 + 0.264i)18-s + 1.05i·19-s + ⋯ |
L(s) = 1 | + (0.982 + 0.187i)2-s − 0.577·3-s + (0.929 + 0.367i)4-s + (−0.567 − 0.108i)6-s + 0.355i·7-s + (0.844 + 0.535i)8-s + 0.333·9-s − 1.35i·11-s + (−0.536 − 0.212i)12-s + 1.54·13-s + (−0.0665 + 0.349i)14-s + (0.729 + 0.683i)16-s + 1.83i·17-s + (0.327 + 0.0623i)18-s + 0.242i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.33801 + 0.650078i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.33801 + 0.650078i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.264i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.941iT - 7T^{2} \) |
| 11 | \( 1 + 4.49iT - 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 - 7.55iT - 17T^{2} \) |
| 19 | \( 1 - 1.05iT - 19T^{2} \) |
| 23 | \( 1 - 1.05iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.49iT - 59T^{2} \) |
| 61 | \( 1 + 8.99iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−10.91801270488003178423596395828, −10.34527958349583002693896307601, −8.616549389813603675800261820499, −8.180836523957290741283502798426, −6.70836859372949431436838469762, −5.99232565697478378944750800785, −5.50911098886993741796080005905, −4.05350888111005769883656762104, −3.33983476215168578230037569483, −1.59591393022744840411065286057,
1.33378189332377377405769252178, 2.86838385467434311690789796059, 4.18420043824646641370016337398, 4.89163429170202900306328328012, 5.91361269277593744391465211957, 6.90908076293703441519075710540, 7.44312380831015834111243783975, 8.989904735524230319021786178609, 10.08989519476829719831793076320, 10.72992306461491445961344029193