| L(s) = 1 | + (−0.281 + 0.959i)2-s + (−1.95 − 1.69i)3-s + (−0.841 − 0.540i)4-s + (−0.0903 + 2.23i)5-s + (2.18 − 1.40i)6-s + (0.748 − 0.341i)7-s + (0.755 − 0.654i)8-s + (0.529 + 3.68i)9-s + (−2.11 − 0.716i)10-s + (4.65 − 1.36i)11-s + (0.730 + 2.48i)12-s + (1.40 + 0.639i)13-s + (0.117 + 0.814i)14-s + (3.97 − 4.22i)15-s + (0.415 + 0.909i)16-s + (2.11 + 3.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 + 0.678i)2-s + (−1.13 − 0.980i)3-s + (−0.420 − 0.270i)4-s + (−0.0403 + 0.999i)5-s + (0.890 − 0.572i)6-s + (0.282 − 0.129i)7-s + (0.267 − 0.231i)8-s + (0.176 + 1.22i)9-s + (−0.669 − 0.226i)10-s + (1.40 − 0.412i)11-s + (0.210 + 0.718i)12-s + (0.388 + 0.177i)13-s + (0.0313 + 0.217i)14-s + (1.02 − 1.09i)15-s + (0.103 + 0.227i)16-s + (0.512 + 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.790290 + 0.235015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790290 + 0.235015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.0903 - 2.23i)T \) |
| 23 | \( 1 + (-4.76 - 0.567i)T \) |
| good | 3 | \( 1 + (1.95 + 1.69i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.748 + 0.341i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.65 + 1.36i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 0.639i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.11 - 3.28i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.15 - 2.66i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.23 + 2.07i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.44 + 6.28i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (4.74 - 0.682i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.536 + 3.73i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.999 - 0.866i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + (-5.78 + 2.64i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.542 + 1.18i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.65 - 9.98i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.59 + 15.6i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.62 + 2.82i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.34 - 5.20i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.95 - 6.47i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-11.8 + 1.69i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.58 + 4.13i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (11.2 + 1.61i)T + (93.0 + 27.3i)T^{2} \) |
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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
−12.09666423620878651489513019849, −11.41099680155241312841893588458, −10.60918873575102805933359950375, −9.322570750686063314757943591878, −7.924021675174704455871180410447, −7.05646535902736720697219246445, −6.31143737691499317467942359757, −5.59171587232047679113295402309, −3.77430339517469811704960991499, −1.32752192571761419320756970927,
1.10765712310247569054748580534, 3.64168408449644120331798266604, 4.81102782795472226207815615631, 5.38851057962622438807014356710, 6.99901212873011609816276636130, 8.705916898654680043246244914859, 9.364482214920976642181291102991, 10.22577012153564249503468474871, 11.37655605579591566856995785893, 11.78691484297810338268407369194
