L(s) = 1 | + (−0.173 − 0.984i)3-s + (0.173 + 0.300i)5-s + (0.939 − 1.62i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)13-s + (0.266 − 0.223i)15-s + 1.53·17-s + (−1.76 − 0.642i)21-s + (0.439 − 0.761i)25-s + (0.5 + 0.866i)27-s + (0.5 + 0.866i)31-s + 0.652·35-s − 0.347·37-s + (0.766 − 0.642i)39-s + (0.766 − 1.32i)43-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)3-s + (0.173 + 0.300i)5-s + (0.939 − 1.62i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)13-s + (0.266 − 0.223i)15-s + 1.53·17-s + (−1.76 − 0.642i)21-s + (0.439 − 0.761i)25-s + (0.5 + 0.866i)27-s + (0.5 + 0.866i)31-s + 0.652·35-s − 0.347·37-s + (0.766 − 0.642i)39-s + (0.766 − 1.32i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.460909071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460909071\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 0.347T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.87T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.223782498217295097326990992817, −7.74161543219726980898370135041, −7.05214633578318266202999216527, −6.55850592848577855801862551540, −5.62975017687288098504407926078, −4.74984429405780828159917966105, −3.91190634924372386119770913778, −2.93563920650582969014263299834, −1.65722199959778919819522399452, −1.03930106763587803135405709687,
1.36867557474791387746263336947, 2.65603964865468624861587147953, 3.33257256759645255391418777525, 4.44570526440947321995955933817, 5.37679043719308304828539637669, 5.48722068134373416429935480604, 6.26614317324476149641010179359, 7.73083928030311701602512651113, 8.263363872179103464570389352578, 8.869782350827012680799101128058