| 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
−14.29110973011615628604248696973, −13.22682901759713579685470763554, −11.95409619088792965079388961454, −11.43776205532966536656007764344, −10.35467703419794476101806286715, −8.632818286926934381724171770173, −7.44271054646394914329590260829, −5.16626792862629083677844663559, −4.70712030966023085814323503615, −2.43115017823415857612742616951,
3.47449961121205404602491674416, 4.76684010630178516127262056569, 6.33404327066658027713502827081, 7.77401047735736018819482841830, 8.255905778588391201980145844999, 10.66355371304693127715420457081, 11.55532425179531719839305185548, 12.51090055910881410979270796630, 13.83040791108862428786894126860, 14.72480652280856824394156252656
