| L(s) = 1 | + (1.09 − 0.891i)2-s + (0.136 − 1.84i)3-s + (0.409 − 1.95i)4-s + (−1.23 + 1.17i)5-s + (−1.49 − 2.15i)6-s + (2.14 + 0.536i)7-s + (−1.29 − 2.51i)8-s + (−0.435 − 0.0646i)9-s + (−0.306 + 2.38i)10-s + (−1.50 − 5.42i)11-s + (−3.56 − 1.02i)12-s + (−4.58 − 1.76i)13-s + (2.82 − 1.32i)14-s + (2.00 + 2.44i)15-s + (−3.66 − 1.60i)16-s + (3.38 + 2.77i)17-s + ⋯ |
| L(s) = 1 | + (0.776 − 0.630i)2-s + (0.0787 − 1.06i)3-s + (0.204 − 0.978i)4-s + (−0.551 + 0.525i)5-s + (−0.612 − 0.878i)6-s + (0.808 + 0.202i)7-s + (−0.458 − 0.888i)8-s + (−0.145 − 0.0215i)9-s + (−0.0969 + 0.755i)10-s + (−0.452 − 1.63i)11-s + (−1.02 − 0.295i)12-s + (−1.27 − 0.490i)13-s + (0.755 − 0.352i)14-s + (0.517 + 0.630i)15-s + (−0.916 − 0.400i)16-s + (0.820 + 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.656929 - 1.93266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.656929 - 1.93266i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.891i)T \) |
| good | 3 | \( 1 + (-0.136 + 1.84i)T + (-2.96 - 0.440i)T^{2} \) |
| 5 | \( 1 + (1.23 - 1.17i)T + (0.245 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-2.14 - 0.536i)T + (6.17 + 3.29i)T^{2} \) |
| 11 | \( 1 + (1.50 + 5.42i)T + (-9.43 + 5.65i)T^{2} \) |
| 13 | \( 1 + (4.58 + 1.76i)T + (9.63 + 8.73i)T^{2} \) |
| 17 | \( 1 + (-3.38 - 2.77i)T + (3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.512 + 2.28i)T + (-17.1 + 8.12i)T^{2} \) |
| 23 | \( 1 + (0.723 - 2.02i)T + (-17.7 - 14.5i)T^{2} \) |
| 29 | \( 1 + (-6.05 - 4.72i)T + (7.04 + 28.1i)T^{2} \) |
| 31 | \( 1 + (-7.72 - 1.53i)T + (28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (3.39 - 0.589i)T + (34.8 - 12.4i)T^{2} \) |
| 41 | \( 1 + (0.473 + 9.63i)T + (-40.8 + 4.01i)T^{2} \) |
| 43 | \( 1 + (-7.02 - 6.06i)T + (6.30 + 42.5i)T^{2} \) |
| 47 | \( 1 + (0.130 + 0.244i)T + (-26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (8.18 - 6.38i)T + (12.8 - 51.4i)T^{2} \) |
| 59 | \( 1 + (-2.07 - 5.37i)T + (-43.7 + 39.6i)T^{2} \) |
| 61 | \( 1 + (-12.2 + 4.03i)T + (48.9 - 36.3i)T^{2} \) |
| 67 | \( 1 + (-8.15 - 4.10i)T + (39.9 + 53.8i)T^{2} \) |
| 71 | \( 1 + (3.68 + 2.73i)T + (20.6 + 67.9i)T^{2} \) |
| 73 | \( 1 + (-15.2 + 3.81i)T + (64.3 - 34.4i)T^{2} \) |
| 79 | \( 1 + (-2.54 - 8.38i)T + (-65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (8.52 + 1.47i)T + (78.1 + 27.9i)T^{2} \) |
| 89 | \( 1 + (3.99 + 11.1i)T + (-68.7 + 56.4i)T^{2} \) |
| 97 | \( 1 + (6.96 - 4.65i)T + (37.1 - 89.6i)T^{2} \) |
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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
−10.85008709965927550748253475357, −10.04445413681396510504895301605, −8.508609448311376217370105193274, −7.74055454569873457670752157202, −6.85455702331448794828632350125, −5.78296419541738737771062105098, −4.86387061587546522532028473721, −3.38616734779114446380184722906, −2.48461101646822809631155617931, −1.00011859075277579618903082090,
2.44205287706340393153813850359, 4.04960480942811760486269256495, 4.75401547818167680522690790482, 4.98718668260316765334057534721, 6.73660957658880428644143146233, 7.69443672777658880597357612566, 8.260041246584771050687219078184, 9.664861823377467047802194519503, 10.12159214130189813481244434250, 11.51461434738038187783140883299
