L(s) = 1 | + (1.20 + 0.439i)3-s + (0.500 + 0.419i)9-s + (1.62 − 0.939i)11-s + (−0.766 + 0.642i)17-s + (−0.984 − 0.173i)19-s + (−0.939 + 0.342i)25-s + (−0.223 − 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (0.984 − 0.173i)43-s + (0.5 + 0.866i)49-s + (−1.20 + 0.439i)51-s + (−1.11 − 0.642i)57-s + (0.524 − 0.439i)59-s + (−1.50 − 1.26i)67-s + ⋯ |
L(s) = 1 | + (1.20 + 0.439i)3-s + (0.500 + 0.419i)9-s + (1.62 − 0.939i)11-s + (−0.766 + 0.642i)17-s + (−0.984 − 0.173i)19-s + (−0.939 + 0.342i)25-s + (−0.223 − 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (0.984 − 0.173i)43-s + (0.5 + 0.866i)49-s + (−1.20 + 0.439i)51-s + (−1.11 − 0.642i)57-s + (0.524 − 0.439i)59-s + (−1.50 − 1.26i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.578772951\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578772951\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
good | 3 | \( 1 + (-1.20 - 0.439i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.524 + 0.439i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (1.50 + 1.26i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−9.664407487662097205072237535755, −9.043402984284204121653523960395, −8.566943861276778336103046211499, −7.78475671458919818679733624034, −6.55669722805347717045226531704, −5.98269373651606472665314733886, −4.39142970610450376511638233786, −3.84997157259933673593529863929, −2.92200061633886908781910854403, −1.69335723270342150790973741657,
1.71730106099862639184364120584, 2.47497633780502933382292621077, 3.79548342097692875841267563506, 4.40205678535720652037113666160, 5.85180875607562226604481835214, 6.93418245425843655548396471840, 7.31506519699841310325507612678, 8.484306293323924849265632649148, 8.938164518796250414156038944089, 9.624970354294502192386385289960