| L(s) = 1 | + (−1.17 − 0.779i)2-s + (−0.159 + 1.72i)3-s + (0.784 + 1.83i)4-s + (−1.58 + 1.57i)5-s + (1.53 − 1.91i)6-s + 1.93·7-s + (0.507 − 2.78i)8-s + (−2.94 − 0.549i)9-s + (3.10 − 0.622i)10-s + (1.02 + 3.14i)11-s + (−3.29 + 1.06i)12-s + (1.21 + 0.395i)13-s + (−2.28 − 1.50i)14-s + (−2.46 − 2.98i)15-s + (−2.76 + 2.88i)16-s + (−5.16 − 3.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.834 − 0.551i)2-s + (−0.0919 + 0.995i)3-s + (0.392 + 0.919i)4-s + (−0.709 + 0.704i)5-s + (0.625 − 0.780i)6-s + 0.730·7-s + (0.179 − 0.983i)8-s + (−0.983 − 0.183i)9-s + (0.980 − 0.196i)10-s + (0.307 + 0.947i)11-s + (−0.951 + 0.306i)12-s + (0.337 + 0.109i)13-s + (−0.609 − 0.402i)14-s + (−0.636 − 0.771i)15-s + (−0.692 + 0.721i)16-s + (−1.25 − 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246013 + 0.516220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246013 + 0.516220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 0.779i)T \) |
| 3 | \( 1 + (0.159 - 1.72i)T \) |
| 5 | \( 1 + (1.58 - 1.57i)T \) |
| good | 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 + (-1.02 - 3.14i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.21 - 0.395i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (5.16 + 3.75i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.96 - 5.45i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.88 - 1.91i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.10 - 4.27i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.595 - 0.818i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (6.78 + 2.20i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-9.94 - 3.23i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 + (-5.17 - 7.12i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.18 + 1.58i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 5.58i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.165 + 0.508i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.39 - 3.19i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (5.19 - 3.77i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.50 + 0.488i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.41 - 3.32i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.93 + 5.41i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-7.71 + 2.50i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.95 - 12.3i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.71281406544043165986468501913, −10.96480831291667018517323906634, −10.37685874090961396553349174965, −9.373430884591395060477699366570, −8.450811668415270912732973762614, −7.55154412455155450119639386136, −6.39276380174638747586065874121, −4.52058233322602522175380412062, −3.77332946538793177406601832035, −2.26739053479656373801894384076,
0.54526620286379786688450446891, 2.08167469792187883497632015166, 4.38666458429745732479083062586, 5.77288320069683538744299240978, 6.63398105364227183538758820047, 7.75473154498340985619256697320, 8.577563035818529023004567687680, 8.820570573795399297533072506247, 10.74283235842435085403170974273, 11.27531915374030253017425552378
