| L(s) = 1 | + (0.100 + 1.41i)2-s + (−4.70 − 1.02i)3-s + (−1.97 + 0.284i)4-s + (4.16 + 2.76i)5-s + (0.969 − 6.74i)6-s + (−4.50 + 8.25i)7-s + (−0.601 − 2.76i)8-s + (12.9 + 5.89i)9-s + (−3.48 + 6.15i)10-s + (5.06 + 5.84i)11-s + (9.60 + 0.687i)12-s + (−9.15 − 16.7i)13-s + (−12.0 − 5.52i)14-s + (−16.7 − 17.2i)15-s + (3.83 − 1.12i)16-s + (−17.1 − 22.9i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (−1.56 − 0.341i)3-s + (−0.494 + 0.0711i)4-s + (0.832 + 0.553i)5-s + (0.161 − 1.12i)6-s + (−0.643 + 1.17i)7-s + (−0.0751 − 0.345i)8-s + (1.43 + 0.655i)9-s + (−0.348 + 0.615i)10-s + (0.460 + 0.531i)11-s + (0.800 + 0.0572i)12-s + (−0.704 − 1.28i)13-s + (−0.863 − 0.394i)14-s + (−1.11 − 1.15i)15-s + (0.239 − 0.0704i)16-s + (−1.00 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0126863 - 0.0189529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0126863 - 0.0189529i\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.16 - 2.76i)T \) |
| 23 | \( 1 + (2.71 + 22.8i)T \) |
| good | 3 | \( 1 + (4.70 + 1.02i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (4.50 - 8.25i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-5.06 - 5.84i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (9.15 + 16.7i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (17.1 + 22.9i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (23.1 - 3.32i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-8.14 - 1.17i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-2.79 - 1.79i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (0.589 + 1.58i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (23.5 + 51.6i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (49.6 + 10.7i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (21.5 - 21.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-56.4 - 30.8i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-16.5 + 56.3i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (41.7 + 26.8i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-4.38 - 61.3i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (66.0 - 76.2i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (43.2 - 57.8i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-36.3 + 123. i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (59.5 - 22.2i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-59.1 - 92.0i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-9.81 - 3.65i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.90186295276299479399483670021, −10.64738533031820665467038658626, −9.870942321296827426729773021756, −8.766966239993188655594374684711, −7.05184629350900019453251762515, −6.48494124892232704008899133164, −5.66523069538441572588499709368, −4.85887377070063250741658017095, −2.49306551312735679559304443991, −0.01451729552769246948291916996,
1.58567009445149947238354276825, 4.03064049576832785011567892560, 4.75172154347417920039282130687, 6.15562342579828679005773226593, 6.70818103307236757117306162792, 8.756254337002303106191122175807, 9.841794719730102285944190916868, 10.41184112896927765955425535596, 11.27344604081916049645586924956, 12.09051786769991633518143338254
