L(s) = 1 | + 2-s + (0.243 + 1.71i)3-s + 4-s + (−2.60 + 1.50i)5-s + (0.243 + 1.71i)6-s + (−2.60 − 0.446i)7-s + 8-s + (−2.88 + 0.833i)9-s + (−2.60 + 1.50i)10-s + (2.01 + 3.49i)11-s + (0.243 + 1.71i)12-s + (0.138 − 3.60i)13-s + (−2.60 − 0.446i)14-s + (−3.21 − 4.09i)15-s + 16-s − 3.34·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.140 + 0.990i)3-s + 0.5·4-s + (−1.16 + 0.672i)5-s + (0.0992 + 0.700i)6-s + (−0.985 − 0.168i)7-s + 0.353·8-s + (−0.960 + 0.277i)9-s + (−0.823 + 0.475i)10-s + (0.607 + 1.05i)11-s + (0.0701 + 0.495i)12-s + (0.0382 − 0.999i)13-s + (−0.696 − 0.119i)14-s + (−0.828 − 1.05i)15-s + 0.250·16-s − 0.810·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.274495 + 1.19608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.274495 + 1.19608i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.243 - 1.71i)T \) |
| 7 | \( 1 + (2.60 + 0.446i)T \) |
| 13 | \( 1 + (-0.138 + 3.60i)T \) |
good | 5 | \( 1 + (2.60 - 1.50i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + (2.56 - 4.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.48iT - 23T^{2} \) |
| 29 | \( 1 + (-3.74 - 2.16i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.95 - 5.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.57iT - 37T^{2} \) |
| 41 | \( 1 + (-6.29 - 3.63i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.23 + 3.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.77 + 3.33i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.50 - 4.90i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3.18iT - 59T^{2} \) |
| 61 | \( 1 + (5.77 + 3.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.04 + 4.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.436 - 0.756i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.78 - 11.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.50 - 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 3.63iT - 89T^{2} \) |
| 97 | \( 1 + (-1.10 - 1.91i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.02726569526331170430248304805, −10.48645223131806839040505021439, −9.649318099329944144587200956934, −8.511271215898637482185835305897, −7.42356436067693190868373602678, −6.63330040084636475800569989186, −5.48518478571649906957303044340, −4.16684399665426166267555294605, −3.72854369227220114195361527820, −2.71290603493578128656284059521,
0.53447847790931191079178504036, 2.44560358358059147470552682389, 3.66071778703495609529526691688, 4.54244343485144679370189415317, 6.09062112630877300533940733458, 6.61650158452540165505535497762, 7.57304683405743307104373499207, 8.698642599697686143478374904803, 9.090542722667254618396420821763, 10.88852301602414561912481207002