| L(s) = 1 | + (0.988 + 0.149i)2-s + (−1.67 + 0.447i)3-s + (0.955 + 0.294i)4-s + (−0.697 − 0.335i)5-s + (−1.72 + 0.193i)6-s + (−1.34 − 2.28i)7-s + (0.900 + 0.433i)8-s + (2.59 − 1.49i)9-s + (−0.639 − 0.435i)10-s + (−3.78 + 4.74i)11-s + (−1.73 − 0.0653i)12-s + (2.78 + 0.419i)13-s + (−0.985 − 2.45i)14-s + (1.31 + 0.249i)15-s + (0.826 + 0.563i)16-s + (7.16 − 2.21i)17-s + ⋯ |
| L(s) = 1 | + (0.699 + 0.105i)2-s + (−0.966 + 0.258i)3-s + (0.477 + 0.147i)4-s + (−0.311 − 0.150i)5-s + (−0.702 + 0.0789i)6-s + (−0.506 − 0.862i)7-s + (0.318 + 0.153i)8-s + (0.866 − 0.499i)9-s + (−0.202 − 0.137i)10-s + (−1.14 + 1.43i)11-s + (−0.499 − 0.0188i)12-s + (0.771 + 0.116i)13-s + (−0.263 − 0.656i)14-s + (0.339 + 0.0644i)15-s + (0.206 + 0.140i)16-s + (1.73 − 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51163 - 0.336259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51163 - 0.336259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.988 - 0.149i)T \) |
| 3 | \( 1 + (1.67 - 0.447i)T \) |
| 7 | \( 1 + (1.34 + 2.28i)T \) |
| good | 5 | \( 1 + (0.697 + 0.335i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (3.78 - 4.74i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.78 - 0.419i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-7.16 + 2.21i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.86 + 4.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.34 + 5.88i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-6.05 - 5.61i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-4.15 + 7.20i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.42 - 4.10i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (0.274 + 3.66i)T + (-40.5 + 6.11i)T^{2} \) |
| 43 | \( 1 + (-0.358 + 4.78i)T + (-42.5 - 6.40i)T^{2} \) |
| 47 | \( 1 + (-3.87 - 0.584i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-3.11 + 2.88i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.248 + 3.31i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (1.65 - 0.509i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (3.20 - 5.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 8.18i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.515 - 1.31i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-1.05 - 1.82i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-15.1 + 2.27i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (8.53 - 1.28i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-2.51 + 4.36i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.19703128810334230536826659293, −9.689634057403585314205145877217, −8.063358798886449895136809301998, −7.25559058373288494554942321796, −6.62198922823221224505462483250, −5.53898239942248882009286498594, −4.71720397613264324445356284852, −4.05693226479996476041104238039, −2.77518906275481825408178123108, −0.803977371939190492056901909309,
1.22246507263083872801865276419, 2.98239579553177956181679601584, 3.70826411867184782598827502643, 5.29699176266941080539204092574, 5.78490026086573561011828325479, 6.24186328223263297767423643444, 7.74885647838265214960721320027, 8.099752752538392072653756482727, 9.670489693841906712800322836940, 10.44171599954121973950768487435
