| L(s) = 1 | + (−0.100 − 1.41i)2-s + (1.39 + 0.302i)3-s + (−1.97 + 0.284i)4-s + (−1.48 − 4.77i)5-s + (0.286 − 1.99i)6-s + (−2.83 + 5.20i)7-s + (0.601 + 2.76i)8-s + (−6.34 − 2.89i)9-s + (−6.58 + 2.58i)10-s + (−12.2 − 14.0i)11-s + (−2.84 − 0.203i)12-s + (2.42 + 4.43i)13-s + (7.62 + 3.48i)14-s + (−0.627 − 7.09i)15-s + (3.83 − 1.12i)16-s + (−2.61 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (0.463 + 0.100i)3-s + (−0.494 + 0.0711i)4-s + (−0.297 − 0.954i)5-s + (0.0477 − 0.332i)6-s + (−0.405 + 0.742i)7-s + (0.0751 + 0.345i)8-s + (−0.704 − 0.321i)9-s + (−0.658 + 0.258i)10-s + (−1.10 − 1.28i)11-s + (−0.236 − 0.0169i)12-s + (0.186 + 0.341i)13-s + (0.544 + 0.248i)14-s + (−0.0418 − 0.472i)15-s + (0.239 − 0.0704i)16-s + (−0.154 − 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0648i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0229245 + 0.706463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0229245 + 0.706463i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (1.48 + 4.77i)T \) |
| 23 | \( 1 + (-9.89 + 20.7i)T \) |
| good | 3 | \( 1 + (-1.39 - 0.302i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (2.83 - 5.20i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (12.2 + 14.0i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-2.42 - 4.43i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (2.61 + 3.49i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (7.14 - 1.02i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (29.0 + 4.18i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-33.9 - 21.8i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (9.01 + 24.1i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (0.860 + 1.88i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-4.77 - 1.03i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-21.6 + 21.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (44.8 + 24.5i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-0.851 + 2.89i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-78.2 - 50.3i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (3.50 + 49.0i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-56.5 + 65.3i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-38.8 + 51.8i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-36.6 + 124. i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-11.3 + 4.21i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (3.24 + 5.04i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-61.9 - 23.1i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.55946386114616136747179981825, −10.63946128832044071673478606157, −9.267216696918572498220609343184, −8.730760531975303494771427812693, −8.036821501313859858517986498273, −6.06353246690576137335823958292, −5.05689999515858754136795209121, −3.57523037359050801148060745337, −2.49407261144185798620686499028, −0.34702516566569030019908286504,
2.56411194672048151333973346956, 3.87716385995138583626450014599, 5.33228422843830161325231673601, 6.66737648236663358380331144537, 7.53387457931652101795070590566, 8.140261855064989959294684191739, 9.602388685465739101758148204831, 10.40222141197042886737870140878, 11.32390949640757087952842574174, 12.82695518089024508180452073915
