| L(s) = 1 | + (1.27 − 0.614i)2-s + (0.306 − 0.944i)3-s + (1.24 − 1.56i)4-s + (−0.152 + 2.23i)5-s + (−0.189 − 1.39i)6-s + (1.38 + 1.38i)7-s + (0.620 − 2.75i)8-s + (1.62 + 1.18i)9-s + (1.17 + 2.93i)10-s + (0.734 − 4.64i)11-s + (−1.09 − 1.65i)12-s + (−0.701 + 0.965i)13-s + (2.61 + 0.912i)14-s + (2.05 + 0.828i)15-s + (−0.906 − 3.89i)16-s + (−0.145 + 0.285i)17-s + ⋯ |
| L(s) = 1 | + (0.900 − 0.434i)2-s + (0.177 − 0.545i)3-s + (0.621 − 0.783i)4-s + (−0.0684 + 0.997i)5-s + (−0.0775 − 0.567i)6-s + (0.523 + 0.523i)7-s + (0.219 − 0.975i)8-s + (0.543 + 0.394i)9-s + (0.372 + 0.928i)10-s + (0.221 − 1.39i)11-s + (−0.316 − 0.477i)12-s + (−0.194 + 0.267i)13-s + (0.699 + 0.243i)14-s + (0.531 + 0.213i)15-s + (−0.226 − 0.973i)16-s + (−0.0352 + 0.0692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33613 - 0.992866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33613 - 0.992866i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 + 0.614i)T \) |
| 5 | \( 1 + (0.152 - 2.23i)T \) |
| good | 3 | \( 1 + (-0.306 + 0.944i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-1.38 - 1.38i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.734 + 4.64i)T + (-10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (0.701 - 0.965i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.145 - 0.285i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (3.33 + 1.69i)T + (11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (0.610 - 3.85i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (4.20 - 2.14i)T + (17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (2.48 - 0.806i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.84 + 3.91i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.63 - 7.75i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.81iT - 43T^{2} \) |
| 47 | \( 1 + (-0.411 - 0.806i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.895 - 2.75i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.704 - 4.44i)T + (-56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (1.08 - 6.86i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 3.36i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.37 + 10.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.821 - 0.130i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.64 - 11.2i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.49 + 4.59i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (11.0 - 8.02i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 5.10i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.20674043582570993276482165828, −10.71802355176716003400162635154, −9.521692480993739870092742733505, −8.214264522858477187287209674656, −7.18304128612964690281957648177, −6.34076625177744299107134099851, −5.38041410189510359756433247926, −4.01082945935291937879048073222, −2.85560829810406590874254367227, −1.75480535809603374752600068409,
1.91719071757374758929450725651, 3.89403581636036042696235048124, 4.43526720901497075300056778508, 5.25287282469442378169242666977, 6.66054032458387177861720859895, 7.57537188058956583986848431899, 8.489067508585486778642826307678, 9.577133107852099795359104494187, 10.47240589398691690366451235188, 11.65689243739187974736400075705
