| L(s) = 1 | + (−0.479 − 0.877i)2-s + (0.0160 + 0.224i)3-s + (−0.540 + 0.841i)4-s + (−1.80 − 1.32i)5-s + (0.189 − 0.121i)6-s + (3.97 + 1.48i)7-s + (0.997 + 0.0713i)8-s + (2.91 − 0.419i)9-s + (−0.293 + 2.21i)10-s + (−0.850 − 2.89i)11-s + (−0.197 − 0.107i)12-s + (−2.04 − 5.49i)13-s + (−0.603 − 4.20i)14-s + (0.267 − 0.425i)15-s + (−0.415 − 0.909i)16-s + (0.313 − 1.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.338 − 0.620i)2-s + (0.00926 + 0.129i)3-s + (−0.270 + 0.420i)4-s + (−0.807 − 0.590i)5-s + (0.0772 − 0.0496i)6-s + (1.50 + 0.560i)7-s + (0.352 + 0.0252i)8-s + (0.973 − 0.139i)9-s + (−0.0928 + 0.700i)10-s + (−0.256 − 0.873i)11-s + (−0.0569 − 0.0311i)12-s + (−0.568 − 1.52i)13-s + (−0.161 − 1.12i)14-s + (0.0689 − 0.109i)15-s + (−0.103 − 0.227i)16-s + (0.0759 − 0.349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874524 - 0.589427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874524 - 0.589427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.479 + 0.877i)T \) |
| 5 | \( 1 + (1.80 + 1.32i)T \) |
| 23 | \( 1 + (2.36 + 4.17i)T \) |
| good | 3 | \( 1 + (-0.0160 - 0.224i)T + (-2.96 + 0.426i)T^{2} \) |
| 7 | \( 1 + (-3.97 - 1.48i)T + (5.29 + 4.58i)T^{2} \) |
| 11 | \( 1 + (0.850 + 2.89i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.04 + 5.49i)T + (-9.82 + 8.51i)T^{2} \) |
| 17 | \( 1 + (-0.313 + 1.44i)T + (-15.4 - 7.06i)T^{2} \) |
| 19 | \( 1 + (-4.78 - 3.07i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.42 - 5.32i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.79 - 2.06i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (5.02 + 3.76i)T + (10.4 + 35.5i)T^{2} \) |
| 41 | \( 1 + (0.610 - 4.24i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.13 + 0.152i)T + (42.5 - 6.11i)T^{2} \) |
| 47 | \( 1 + (-1.03 - 1.03i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.92 - 7.85i)T + (-40.0 - 34.7i)T^{2} \) |
| 59 | \( 1 + (9.55 + 4.36i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.07 - 6.99i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.85 - 2.10i)T + (36.2 - 56.3i)T^{2} \) |
| 71 | \( 1 + (-7.76 - 2.27i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (5.59 - 1.21i)T + (66.4 - 30.3i)T^{2} \) |
| 79 | \( 1 + (-4.78 + 10.4i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.282 - 0.376i)T + (-23.3 - 79.6i)T^{2} \) |
| 89 | \( 1 + (6.58 - 7.59i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-7.75 - 10.3i)T + (-27.3 + 93.0i)T^{2} \) |
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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
−12.20057651320404518842382292489, −11.02467827958922442821050477698, −10.30853706641178945142801947654, −8.997413138000801455672298682180, −8.087516406126705935189633288759, −7.58727726230471810950314535252, −5.39807465544773153408733722403, −4.58348107781735545315746498881, −3.12239674869975339078746592607, −1.18507008665096823616828974533,
1.77650328663794624386311328175, 4.21864422820462199558789673569, 4.85071333337091629569430025753, 6.79057590515737294535484421192, 7.45148372234035784059680519090, 8.009841918285295470401931740436, 9.486490641599555637938039803892, 10.40113889952118082232244074050, 11.45135423668716493120460850280, 12.11228123724315168138894790507
