L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.173 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (−0.642 + 0.766i)6-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + 12-s + (0.424 + 1.58i)13-s + (−0.939 − 0.342i)16-s + (0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.642 − 0.766i)24-s + (−0.866 + 0.5i)25-s + (0.939 − 1.34i)26-s + (0.5 + 0.866i)27-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.173 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (−0.642 + 0.766i)6-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + 12-s + (0.424 + 1.58i)13-s + (−0.939 − 0.342i)16-s + (0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.642 − 0.766i)24-s + (−0.866 + 0.5i)25-s + (0.939 − 1.34i)26-s + (0.5 + 0.866i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6699014257\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6699014257\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-0.424 - 1.58i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (0.0451 - 0.168i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.342 - 0.592i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 1.28iT - T^{2} \) |
| 73 | \( 1 - 1.53iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 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
−8.782133337351247659799913077741, −8.109933798068787377513600753025, −7.46063065933997924389623910957, −6.64093927836203151833464302290, −6.09976800814098445815586169319, −4.78409873131255446629879376247, −3.97705395478604783705498313516, −2.89485776933964944602895882256, −2.00113288517309157491406391223, −1.22512577691904591397740350150,
0.56029304836549434446870244296, 2.25496473793740403850289740773, 3.49568848160537608051219232732, 4.32067373089683628313788704777, 5.31052435136476236241437824360, 5.77333446759340898508103992055, 6.44712985092744679080157945123, 7.57214334623243522315296196867, 8.214731627080430518628112811466, 8.692002345517383536053354983941