L(s) = 1 | + 0.823i·2-s + (1.33 − 2.30i)3-s + 1.32·4-s + (−2.73 − 1.58i)5-s + (1.89 + 1.09i)6-s + 2.73i·8-s + (−2.03 − 3.53i)9-s + (1.30 − 2.25i)10-s + (−5.14 − 2.97i)11-s + (1.75 − 3.04i)12-s + (0.0766 − 3.60i)13-s + (−7.28 + 4.20i)15-s + 0.390·16-s − 2.69·17-s + (2.90 − 1.67i)18-s + (1.69 − 0.978i)19-s + ⋯ |
L(s) = 1 | + 0.582i·2-s + (0.767 − 1.33i)3-s + 0.660·4-s + (−1.22 − 0.707i)5-s + (0.774 + 0.447i)6-s + 0.967i·8-s + (−0.679 − 1.17i)9-s + (0.411 − 0.713i)10-s + (−1.55 − 0.895i)11-s + (0.507 − 0.879i)12-s + (0.0212 − 0.999i)13-s + (−1.88 + 1.08i)15-s + 0.0976·16-s − 0.654·17-s + (0.685 − 0.395i)18-s + (0.388 − 0.224i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.889918 - 1.21209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.889918 - 1.21209i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-0.0766 + 3.60i)T \) |
good | 2 | \( 1 - 0.823iT - 2T^{2} \) |
| 3 | \( 1 + (-1.33 + 2.30i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.73 + 1.58i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.14 + 2.97i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + (-1.69 + 0.978i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + (-2.99 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.997 + 0.575i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.50iT - 37T^{2} \) |
| 41 | \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.49 + 6.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.394 - 0.228i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.199 + 0.345i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.80iT - 59T^{2} \) |
| 61 | \( 1 + (0.578 + 1.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.43 - 3.13i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.19 - 4.15i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 6.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.19iT - 83T^{2} \) |
| 89 | \( 1 + 3.56iT - 89T^{2} \) |
| 97 | \( 1 + (-2.96 - 1.71i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.59536069997080740596095449124, −8.645688502965525237835995001315, −8.435273193380735277719271514780, −7.47988354001781969962721326097, −7.32091847360063130320425239285, −5.95663458703174565100471480946, −5.02936967835365309221759360656, −3.27655887040785924003661459237, −2.45010953959591611420510335294, −0.71111397246126526966327570571,
2.40885385650061136543442387225, 3.10627473719498808888576967346, 4.09016843384854143067181245782, 4.80678254286645028307165682800, 6.57684424436489311073584729920, 7.51466693209386793035668543423, 8.184464151505936009255913926174, 9.447776072808326725678789720311, 10.09563009452748901431733278778, 10.82079884594473709387254095606