| L(s) = 1 | + (0.133 + 1.40i)2-s + (0.00307 − 1.73i)3-s + (−1.96 + 0.376i)4-s + (−2.47 − 0.901i)5-s + (2.43 − 0.227i)6-s + (−2.55 − 0.451i)7-s + (−0.793 − 2.71i)8-s + (−2.99 − 0.0106i)9-s + (0.937 − 3.60i)10-s + (0.556 + 1.52i)11-s + (0.646 + 3.40i)12-s + (−1.88 − 2.24i)13-s + (0.292 − 3.66i)14-s + (−1.56 + 4.28i)15-s + (3.71 − 1.48i)16-s + (−3.28 + 1.89i)17-s + ⋯ |
| L(s) = 1 | + (0.0946 + 0.995i)2-s + (0.00177 − 0.999i)3-s + (−0.982 + 0.188i)4-s + (−1.10 − 0.403i)5-s + (0.995 − 0.0928i)6-s + (−0.967 − 0.170i)7-s + (−0.280 − 0.959i)8-s + (−0.999 − 0.00355i)9-s + (0.296 − 1.14i)10-s + (0.167 + 0.461i)11-s + (0.186 + 0.982i)12-s + (−0.522 − 0.622i)13-s + (0.0782 − 0.979i)14-s + (−0.405 + 1.10i)15-s + (0.928 − 0.370i)16-s + (−0.797 + 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.188949 - 0.310442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.188949 - 0.310442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.133 - 1.40i)T \) |
| 3 | \( 1 + (-0.00307 + 1.73i)T \) |
| good | 5 | \( 1 + (2.47 + 0.901i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.55 + 0.451i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.556 - 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.88 + 2.24i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.28 - 1.89i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.30 + 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 6.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.88 + 3.25i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.57 + 0.630i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-6.40 + 3.69i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.43 + 5.28i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.77 - 1.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.253 - 1.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.180T + 53T^{2} \) |
| 59 | \( 1 + (-0.253 + 0.695i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 8.82i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.57 + 4.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 + 8.01i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.39 + 8.81i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.211 + 0.121i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.95 + 0.711i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.28376258034514064157443980167, −11.39521423436962609003844145672, −9.649580915447459286071558323331, −8.758172960138638998434321775499, −7.57842619697089123352685772478, −7.19117224496761449358912530097, −6.01733578195995992745258354787, −4.68577828036085285568407491266, −3.25571939963486951461969695082, −0.29163749752019806485250920561,
2.95146280993181150950201868877, 3.74231867435470095395764534552, 4.80162813320532267090009053676, 6.29223102144053451950859832075, 7.975539600756411640554986280237, 9.074811844154732714830190131681, 9.852655277655191819413789857977, 10.74972522192365210312438293214, 11.67251782569274386087597904994, 12.18098494206054047396350914622
