| L(s) = 1 | + (−1.02 + 1.46i)3-s + (0.710 − 2.12i)5-s + (−2.65 + 0.711i)7-s + (−0.0626 − 0.172i)9-s + (2.84 + 4.91i)11-s + (−1.87 + 1.31i)13-s + (2.37 + 3.20i)15-s + (1.57 + 3.38i)17-s + (−3.00 + 3.15i)19-s + (1.67 − 4.60i)21-s + (−3.36 + 0.294i)23-s + (−3.98 − 3.01i)25-s + (−4.85 − 1.30i)27-s + (−1.09 + 0.397i)29-s + (7.63 + 4.40i)31-s + ⋯ |
| L(s) = 1 | + (−0.590 + 0.843i)3-s + (0.317 − 0.948i)5-s + (−1.00 + 0.268i)7-s + (−0.0208 − 0.0573i)9-s + (0.856 + 1.48i)11-s + (−0.520 + 0.364i)13-s + (0.612 + 0.828i)15-s + (0.383 + 0.821i)17-s + (−0.688 + 0.724i)19-s + (0.365 − 1.00i)21-s + (−0.701 + 0.0613i)23-s + (−0.797 − 0.602i)25-s + (−0.934 − 0.250i)27-s + (−0.202 + 0.0738i)29-s + (1.37 + 0.791i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.478948 + 0.720137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.478948 + 0.720137i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.710 + 2.12i)T \) |
| 19 | \( 1 + (3.00 - 3.15i)T \) |
| good | 3 | \( 1 + (1.02 - 1.46i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (2.65 - 0.711i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.84 - 4.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.87 - 1.31i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.57 - 3.38i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (3.36 - 0.294i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (1.09 - 0.397i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.63 - 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.85 + 3.85i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.15 - 0.379i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.126 - 1.44i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.0279 - 0.0130i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.107 - 1.23i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-6.00 - 2.18i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.16 - 6.85i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.85 + 12.5i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (7.74 + 9.23i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-10.1 - 7.08i)T + (24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (2.15 + 12.2i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.01 - 11.2i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.92 + 10.9i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (10.5 - 4.90i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.87908218938353398110164334800, −10.38124881907858099153773438895, −9.838932320895754165852520418189, −9.270207441466392286173224344921, −8.051902717712391398834257398817, −6.65422689153214200186799906953, −5.77381424641051426227586584204, −4.65769625539116730527932815027, −3.97429651301759765680925191660, −1.92463692056796038262682787738,
0.61755309218295755181859545709, 2.66261243255107460633512707872, 3.75338700392175617209740047143, 5.66315789786761989738965322188, 6.49910991122169305565003475672, 6.85757866994547765601872942126, 8.096167112423166493597137078427, 9.456705075022835690238884589680, 10.14220357099175432318970118173, 11.34427277185883237848605805990
