| L(s) = 1 | + (0.997 + 0.0713i)2-s + (0.212 + 0.977i)3-s + (0.989 + 0.142i)4-s + (0.0285 − 2.23i)5-s + (0.142 + 0.989i)6-s + (−1.81 − 3.33i)7-s + (0.977 + 0.212i)8-s + (−0.909 + 0.415i)9-s + (0.188 − 2.22i)10-s + (−0.824 − 0.714i)11-s + (0.0713 + 0.997i)12-s + (1.85 + 1.01i)13-s + (−1.57 − 3.45i)14-s + (2.19 − 0.447i)15-s + (0.959 + 0.281i)16-s + (4.77 − 6.37i)17-s + ⋯ |
| L(s) = 1 | + (0.705 + 0.0504i)2-s + (0.122 + 0.564i)3-s + (0.494 + 0.0711i)4-s + (0.0127 − 0.999i)5-s + (0.0580 + 0.404i)6-s + (−0.687 − 1.25i)7-s + (0.345 + 0.0751i)8-s + (−0.303 + 0.138i)9-s + (0.0594 − 0.704i)10-s + (−0.248 − 0.215i)11-s + (0.0205 + 0.287i)12-s + (0.514 + 0.280i)13-s + (−0.421 − 0.922i)14-s + (0.565 − 0.115i)15-s + (0.239 + 0.0704i)16-s + (1.15 − 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96647 - 0.939767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96647 - 0.939767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 - 0.0713i)T \) |
| 3 | \( 1 + (-0.212 - 0.977i)T \) |
| 5 | \( 1 + (-0.0285 + 2.23i)T \) |
| 23 | \( 1 + (1.64 - 4.50i)T \) |
| good | 7 | \( 1 + (1.81 + 3.33i)T + (-3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (0.824 + 0.714i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.85 - 1.01i)T + (7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (-4.77 + 6.37i)T + (-4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.435 + 3.03i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-3.03 + 0.436i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.68 + 3.01i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.45 - 3.91i)T + (-27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (3.28 - 7.19i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (7.16 - 1.55i)T + (39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (-2.60 + 2.60i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.56 + 4.67i)T + (28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.33 - 7.94i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.89 - 4.51i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.00898 - 0.125i)T + (-66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (-4.04 - 4.67i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.31 + 2.47i)T + (20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (14.2 - 4.16i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.10 + 0.411i)T + (62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.56 - 1.00i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.93 - 2.58i)T + (73.3 - 63.5i)T^{2} \) |
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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
−10.06342820266184831184712447924, −9.818518762658500445360374532064, −8.623138268176954014770427226363, −7.65481075565465759538657710795, −6.75951297793609889570051589630, −5.57209208617800320039078847600, −4.77215177064550890067176642009, −3.89726486800266967160737273348, −2.98850854609190719037658582621, −0.947488164410924271874879725463,
1.95610905242666198767199923700, 2.96332831335379091262616939522, 3.75552550145504379897538439360, 5.51200929460749520160763680461, 6.09784387690499537867275985535, 6.77388482374768230632128056380, 7.939253258223840871013312271245, 8.651140326432982773329493874164, 10.08698952337756945243819262514, 10.49669524904786198061465188201
