L(s) = 1 | + (−0.718 − 2.21i)2-s + (−1.86 + 1.35i)3-s + (−2.75 + 2.00i)4-s + (4.33 + 3.14i)6-s − 3.59·7-s + (2.65 + 1.92i)8-s + (0.710 − 2.18i)9-s + (−0.153 − 0.473i)11-s + (2.42 − 7.46i)12-s + (−0.818 + 2.51i)13-s + (2.58 + 7.95i)14-s + (0.250 − 0.771i)16-s + (4.13 + 3.00i)17-s − 5.35·18-s + (0.798 + 0.580i)19-s + ⋯ |
L(s) = 1 | + (−0.508 − 1.56i)2-s + (−1.07 + 0.781i)3-s + (−1.37 + 1.00i)4-s + (1.76 + 1.28i)6-s − 1.35·7-s + (0.938 + 0.681i)8-s + (0.236 − 0.729i)9-s + (−0.0463 − 0.142i)11-s + (0.700 − 2.15i)12-s + (−0.226 + 0.698i)13-s + (0.690 + 2.12i)14-s + (0.0627 − 0.192i)16-s + (1.00 + 0.728i)17-s − 1.26·18-s + (0.183 + 0.133i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.289667 - 0.335583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.289667 - 0.335583i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.718 + 2.21i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.86 - 1.35i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 + (0.153 + 0.473i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.818 - 2.51i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.13 - 3.00i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.798 - 0.580i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.98 + 6.09i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.50 + 3.27i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.89 + 3.55i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.42 + 4.37i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.888 - 2.73i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 9.48T + 43T^{2} \) |
| 47 | \( 1 + (-4.34 + 3.15i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.248 - 0.180i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.391 - 1.20i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.92 + 5.92i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (4.27 + 3.10i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.122 + 0.0893i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.59 - 14.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-13.4 + 9.75i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.8 - 8.57i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.51 - 10.8i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.687 + 0.499i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.53464412852672520614318927684, −9.714461199203826488688974834190, −9.333719147164002140563703420815, −8.081271280992999128976894754185, −6.52805760550189782968373890161, −5.74205135405605082174484614607, −4.34968282069563998708424365312, −3.64094241610560098932926303500, −2.40763538325450860008928193306, −0.52374286363108390194574204620,
0.77863013848943149582561579523, 3.21551117405045662126972490302, 5.10750564846443115560656364740, 5.76564803673611194712482770491, 6.41926208127287020411785347010, 7.26681650399969752644119914587, 7.67614627484377190320525785107, 9.054973195538249007446893069008, 9.709626653346970962683763426544, 10.61956172528841008436233105307