| L(s) = 1 | + (−0.342 + 0.939i)2-s + (0.288 + 1.70i)3-s + (−0.766 − 0.642i)4-s + (−0.675 + 3.83i)5-s + (−1.70 − 0.312i)6-s + (−0.478 − 2.60i)7-s + (0.866 − 0.500i)8-s + (−2.83 + 0.986i)9-s + (−3.37 − 1.94i)10-s + (−2.70 + 0.477i)11-s + (0.876 − 1.49i)12-s + (0.574 + 1.57i)13-s + (2.60 + 0.440i)14-s + (−6.74 − 0.0466i)15-s + (0.173 + 0.984i)16-s + (−2.48 + 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (0.166 + 0.985i)3-s + (−0.383 − 0.321i)4-s + (−0.302 + 1.71i)5-s + (−0.695 − 0.127i)6-s + (−0.180 − 0.983i)7-s + (0.306 − 0.176i)8-s + (−0.944 + 0.328i)9-s + (−1.06 − 0.615i)10-s + (−0.816 + 0.143i)11-s + (0.252 − 0.431i)12-s + (0.159 + 0.437i)13-s + (0.697 + 0.117i)14-s + (−1.74 − 0.0120i)15-s + (0.0434 + 0.246i)16-s + (−0.602 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.159428 - 0.799572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159428 - 0.799572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 3 | \( 1 + (-0.288 - 1.70i)T \) |
| 7 | \( 1 + (0.478 + 2.60i)T \) |
| good | 5 | \( 1 + (0.675 - 3.83i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (2.70 - 0.477i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.574 - 1.57i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.48 - 4.30i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.24 + 3.60i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.43 + 4.09i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.869 - 2.38i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (5.22 - 6.22i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.177 - 0.306i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.97 + 1.44i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.65 - 9.39i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.79 - 4.85i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 6.43iT - 53T^{2} \) |
| 59 | \( 1 + (0.927 - 5.26i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.29 - 3.93i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.47 - 0.537i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.90 - 3.98i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.45 - 0.842i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-15.6 - 5.67i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.91 - 1.42i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.32 - 4.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.44 - 1.48i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.21181409397761364802217454925, −10.83345766984944312943154260895, −10.16603617155581128319503237889, −9.238799348214711555212702769495, −8.003151490529456342276925995407, −7.14928659887084574463831588923, −6.41589256675146299176309298789, −5.01341590230992849580616077271, −3.84416554480473933737354683802, −2.87719890327348718431552341417,
0.57439192843461687278393647401, 2.01464028947751825518688168739, 3.36262334980629191748465021820, 5.12195990193721857012257317739, 5.66563033597735879361505005867, 7.53063521443215574721287574657, 8.133783886150777346102838101922, 9.083300234129871899574325438122, 9.519976349945393975211037957300, 11.27302632386673034183281953879
