| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.0812 − 0.460i)5-s + (−0.939 + 0.342i)6-s + (2.20 − 3.82i)7-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.358 − 0.300i)10-s + (2.76 + 4.79i)11-s + (−0.499 + 0.866i)12-s + (−5.62 + 2.04i)13-s + (−0.766 − 4.34i)14-s + (−0.0812 + 0.460i)15-s + (−0.939 − 0.342i)16-s + (−3.33 + 2.79i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0363 − 0.206i)5-s + (−0.383 + 0.139i)6-s + (0.833 − 1.44i)7-s + (−0.176 − 0.306i)8-s + (0.255 + 0.214i)9-s + (−0.113 − 0.0951i)10-s + (0.833 + 1.44i)11-s + (−0.144 + 0.250i)12-s + (−1.55 + 0.567i)13-s + (−0.204 − 1.16i)14-s + (−0.0209 + 0.118i)15-s + (−0.234 − 0.0855i)16-s + (−0.807 + 0.677i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01572 - 0.662171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01572 - 0.662171i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-4.34 - 0.405i)T \) |
| good | 5 | \( 1 + (0.0812 + 0.460i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.20 + 3.82i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.76 - 4.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.62 - 2.04i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.33 - 2.79i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.549 - 3.11i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.15 + 0.970i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 1.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 + (-1.84 - 0.669i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.624 - 3.54i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.08 + 5.94i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.464 - 2.63i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.01 + 0.853i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.15 - 6.53i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.90 + 1.60i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.31 + 7.48i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.97 + 1.08i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.18 + 0.432i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.96 + 15.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.7 - 4.26i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (6.36 - 5.34i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.32015367496440281128589436059, −12.22959509098545717662517574002, −11.54747820549591425117988292824, −10.41441476216714459728155508945, −9.530148708633396892246165806370, −7.53738899281341391381666165037, −6.80229014550440123008964503563, −4.88378225940697214772082102543, −4.24712382518533956421844402942, −1.67591113988167962295001015136,
2.87768604156883120626606761591, 4.87480381338100059325188897645, 5.63968843541801313064519084092, 6.91554202314556867885778213394, 8.352072362910469790878172340043, 9.351414424320660227171213238655, 11.07511080638865597759813251903, 11.76908947835605642864718887502, 12.57001406126703596779944643171, 14.06667155102947930129216456532
