L(s) = 1 | + (−1.70 + 0.984i)7-s + (−0.266 − 1.50i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s − 1.87·37-s + (0.439 − 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (0.673 − 1.85i)67-s + (−0.266 + 1.50i)73-s + (−1.93 − 0.342i)79-s + (1.93 + 2.31i)91-s + (−0.939 + 0.342i)97-s + (−1.11 − 0.642i)103-s + ⋯ |
L(s) = 1 | + (−1.70 + 0.984i)7-s + (−0.266 − 1.50i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s − 1.87·37-s + (0.439 − 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (0.673 − 1.85i)67-s + (−0.266 + 1.50i)73-s + (−1.93 − 0.342i)79-s + (1.93 + 2.31i)91-s + (−0.939 + 0.342i)97-s + (−1.11 − 0.642i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5272807407\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5272807407\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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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
−8.750281116234435246097977540894, −8.223691025311362736664985112590, −7.09385269860957308973443972037, −6.47524183384080472555806971250, −5.74303968164002686555072796897, −5.07471295752938278995300378427, −3.79994411513703291413319344133, −2.96589084186042026135142432250, −2.35739860201720531077296559279, −0.32742209667497723651042750906,
1.48433857041866618193387588187, 2.79863134501186826309233371785, 3.77109988308146855970313278162, 4.22458089905394565061156896591, 5.46842488632759760847589422534, 6.42091143486856957842009071032, 6.88523088785183660164639558418, 7.47909079964892840427089321788, 8.657288753045057223104346096671, 9.278988447776399316546325451914