| L(s) = 1 | + (−0.786 + 0.618i)2-s + (−2.27 + 1.61i)3-s + (0.235 − 0.971i)4-s + (−0.350 − 0.0675i)5-s + (0.786 − 2.67i)6-s + (0.948 − 2.47i)7-s + (0.415 + 0.909i)8-s + (1.56 − 4.52i)9-s + (0.317 − 0.163i)10-s + (3.17 − 4.03i)11-s + (1.03 + 2.59i)12-s + (−2.49 + 3.88i)13-s + (0.781 + 2.52i)14-s + (0.905 − 0.413i)15-s + (−0.888 − 0.458i)16-s + (0.833 − 0.794i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 + 0.437i)2-s + (−1.31 + 0.934i)3-s + (0.117 − 0.485i)4-s + (−0.156 − 0.0301i)5-s + (0.320 − 1.09i)6-s + (0.358 − 0.933i)7-s + (0.146 + 0.321i)8-s + (0.521 − 1.50i)9-s + (0.100 − 0.0516i)10-s + (0.957 − 1.21i)11-s + (0.299 + 0.747i)12-s + (−0.692 + 1.07i)13-s + (0.208 + 0.675i)14-s + (0.233 − 0.106i)15-s + (−0.222 − 0.114i)16-s + (0.202 − 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.636253 + 0.174952i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636253 + 0.174952i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (-0.948 + 2.47i)T \) |
| 23 | \( 1 + (-4.03 - 2.58i)T \) |
| good | 3 | \( 1 + (2.27 - 1.61i)T + (0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (0.350 + 0.0675i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-3.17 + 4.03i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.49 - 3.88i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.833 + 0.794i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.33 - 3.17i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-9.51 - 2.79i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.709 + 7.42i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.49 + 1.21i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-7.03 - 6.09i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.52 - 1.15i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-10.2 + 5.91i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.676 - 0.0322i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.13 + 2.19i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-0.777 + 1.09i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-1.76 + 4.41i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.71 + 11.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.38 - 0.820i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-1.49 - 0.0712i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-2.18 - 2.51i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (10.3 - 0.990i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-3.76 + 4.35i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.50895545987565551593891639078, −10.77910916416869545772472184547, −9.883741088515541183419424054530, −9.163752618358013150338616800419, −7.81639579062241104766993906266, −6.72307474144194263121770096973, −5.86822670811727941805182773119, −4.76128541890061485477691186489, −3.83110493469903416583365954232, −0.892493354442510973041408851151,
1.11668978098205943350986672061, 2.57251980992936687280349210305, 4.69983188897895971590762905831, 5.68254580043171141974711408304, 6.88336735657050690691780024650, 7.52344596180648693212558267949, 8.741703447769203357869187644566, 9.846999084754849114689545105446, 10.81925450776174595541533517143, 11.71786783676797728226266953724
