| L(s) = 1 | + (1.09 + 0.888i)2-s + (−0.220 − 0.185i)3-s + (0.419 + 1.95i)4-s + (−2.14 + 0.779i)5-s + (−0.0781 − 0.399i)6-s + (3.55 − 2.04i)7-s + (−1.27 + 2.52i)8-s + (−0.506 − 2.87i)9-s + (−3.04 − 1.04i)10-s + (−3.61 − 2.08i)11-s + (0.269 − 0.509i)12-s + (−0.374 − 0.446i)13-s + (5.72 + 0.901i)14-s + (0.616 + 0.224i)15-s + (−3.64 + 1.64i)16-s + (−0.573 + 3.25i)17-s + ⋯ |
| L(s) = 1 | + (0.777 + 0.628i)2-s + (−0.127 − 0.106i)3-s + (0.209 + 0.977i)4-s + (−0.957 + 0.348i)5-s + (−0.0318 − 0.163i)6-s + (1.34 − 0.774i)7-s + (−0.451 + 0.892i)8-s + (−0.168 − 0.957i)9-s + (−0.963 − 0.330i)10-s + (−1.09 − 0.630i)11-s + (0.0778 − 0.146i)12-s + (−0.103 − 0.123i)13-s + (1.53 + 0.240i)14-s + (0.159 + 0.0579i)15-s + (−0.911 + 0.410i)16-s + (−0.138 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11084 + 0.480902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11084 + 0.480902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 - 0.888i)T \) |
| 19 | \( 1 + (-0.458 - 4.33i)T \) |
| good | 3 | \( 1 + (0.220 + 0.185i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (2.14 - 0.779i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.55 + 2.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.61 + 2.08i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.374 + 0.446i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.573 - 3.25i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.862 + 2.37i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.34 + 1.47i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.386 - 0.670i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.23iT - 37T^{2} \) |
| 41 | \( 1 + (4.51 - 5.37i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.55 - 4.27i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (4.84 - 0.854i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.232 - 0.639i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.368 + 2.08i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.91 + 1.05i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.44 + 13.8i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 4.51i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (8.59 + 7.20i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.91 - 6.63i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.29 - 0.747i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.52 - 11.3i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 1.78i)T + (91.1 + 33.1i)T^{2} \) |
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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
−14.72480652280856824394156252656, −13.83040791108862428786894126860, −12.51090055910881410979270796630, −11.55532425179531719839305185548, −10.66355371304693127715420457081, −8.255905778588391201980145844999, −7.77401047735736018819482841830, −6.33404327066658027713502827081, −4.76684010630178516127262056569, −3.47449961121205404602491674416,
2.43115017823415857612742616951, 4.70712030966023085814323503615, 5.16626792862629083677844663559, 7.44271054646394914329590260829, 8.632818286926934381724171770173, 10.35467703419794476101806286715, 11.43776205532966536656007764344, 11.95409619088792965079388961454, 13.22682901759713579685470763554, 14.29110973011615628604248696973
