| L(s) = 1 | + (0.603 − 1.27i)2-s + (1.98 + 1.06i)3-s + (−1.27 − 1.54i)4-s + (−0.212 + 0.259i)5-s + (2.55 − 1.89i)6-s + (−0.792 − 0.529i)7-s + (−2.74 + 0.694i)8-s + (1.14 + 1.71i)9-s + (0.203 + 0.428i)10-s + (2.34 + 0.709i)11-s + (−0.886 − 4.41i)12-s + (−2.81 + 2.30i)13-s + (−1.15 + 0.693i)14-s + (−0.696 + 0.288i)15-s + (−0.766 + 3.92i)16-s + (−1.82 − 0.756i)17-s + ⋯ |
| L(s) = 1 | + (0.426 − 0.904i)2-s + (1.14 + 0.612i)3-s + (−0.635 − 0.771i)4-s + (−0.0950 + 0.115i)5-s + (1.04 − 0.775i)6-s + (−0.299 − 0.200i)7-s + (−0.969 + 0.245i)8-s + (0.382 + 0.573i)9-s + (0.0641 + 0.135i)10-s + (0.705 + 0.214i)11-s + (−0.255 − 1.27i)12-s + (−0.779 + 0.640i)13-s + (−0.308 + 0.185i)14-s + (−0.179 + 0.0745i)15-s + (−0.191 + 0.981i)16-s + (−0.442 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45536 - 0.597542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45536 - 0.597542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.603 + 1.27i)T \) |
| good | 3 | \( 1 + (-1.98 - 1.06i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (0.212 - 0.259i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (0.792 + 0.529i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.34 - 0.709i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (2.81 - 2.30i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (1.82 + 0.756i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.157 + 1.60i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (7.27 + 1.44i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.10 - 3.64i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-7.16 + 7.16i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.968 + 0.0953i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (2.34 - 11.8i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-7.65 + 4.09i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.74 - 4.20i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.50 + 8.26i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (2.07 + 1.69i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-3.63 + 6.80i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-5.45 + 10.1i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-1.79 + 2.68i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.04 + 1.36i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.58 - 11.0i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (7.11 + 0.700i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (13.6 - 2.71i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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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
−13.38060326874019966843054473905, −12.17164575850935274967859368703, −11.22479497095736126491332701514, −9.775346433633423514124290137508, −9.499817961896115880718057333951, −8.247556907801319429593014282121, −6.53154013924849264406527978204, −4.63377673567862642252277302680, −3.68576079423547044827991203592, −2.37054950194748650624241150838,
2.73245226033875988404517243887, 4.16468891146697493318092592732, 5.87984096326876587963749272519, 7.06895096284082919626259469921, 8.098027034895520802424645522991, 8.768277508092425281253949398167, 9.976818165117118138070720093264, 12.04323457346639423123344997273, 12.65812819817824719794984719292, 13.89429350761274487378094730068
