| L(s) = 1 | + (−0.639 + 1.26i)2-s + (−1.29 + 3.11i)3-s + (−1.18 − 1.61i)4-s + (−0.453 + 0.891i)5-s + (−3.10 − 3.62i)6-s + (−0.939 − 3.91i)7-s + (2.79 − 0.459i)8-s + (−5.92 − 5.92i)9-s + (−0.833 − 1.14i)10-s + (−3.02 + 3.54i)11-s + (6.55 − 1.60i)12-s + (−0.291 − 0.475i)13-s + (5.53 + 1.31i)14-s + (−2.19 − 2.56i)15-s + (−1.20 + 3.81i)16-s + (−0.291 − 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.452 + 0.891i)2-s + (−0.745 + 1.79i)3-s + (−0.591 − 0.806i)4-s + (−0.203 + 0.398i)5-s + (−1.26 − 1.47i)6-s + (−0.355 − 1.47i)7-s + (0.986 − 0.162i)8-s + (−1.97 − 1.97i)9-s + (−0.263 − 0.361i)10-s + (−0.912 + 1.06i)11-s + (1.89 − 0.462i)12-s + (−0.0808 − 0.131i)13-s + (1.48 + 0.352i)14-s + (−0.565 − 0.662i)15-s + (−0.301 + 0.953i)16-s + (−0.0705 − 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.414295 + 0.164575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414295 + 0.164575i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.639 - 1.26i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (-5.44 + 3.37i)T \) |
| good | 3 | \( 1 + (1.29 - 3.11i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (0.939 + 3.91i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (3.02 - 3.54i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.291 + 0.475i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.291 + 3.69i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-2.15 - 1.32i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-4.80 - 3.49i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0769 + 0.977i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (1.53 - 4.72i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.83 + 8.71i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.945 + 5.96i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (1.69 - 7.04i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-5.93 - 0.467i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 3.12i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.426 - 2.69i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 1.82i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (4.92 + 4.20i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-1.00 + 1.00i)T - 73iT^{2} \) |
| 79 | \( 1 + (15.5 + 6.42i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 7.38iT - 83T^{2} \) |
| 89 | \( 1 + (-8.26 + 1.98i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 8.65i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.33543587622241818609221304630, −9.650277302064452671297569957657, −8.952933675501844264852768741389, −7.44188258632244771649496834128, −7.09334253002181354710458821562, −5.78142759148158206506036414670, −4.98584212986547042843840557296, −4.30166848197537383022872838332, −3.33126584210991080031959377424, −0.36459825702973495206603331316,
0.975246134547517814076856108627, 2.28212317766333205484926370475, 2.99877630181380673836729003695, 5.04984957866164155995688278007, 5.74878915079980050139404198744, 6.69316041817075378159916447349, 7.85055168150000891227155123655, 8.410880343911865574966106977797, 8.995812931173366740701298023873, 10.40698573843819933099795983697
