L(s) = 1 | + (−1.13 − 1.31i)3-s + (2.14 + 0.633i)5-s + (0.907 − 0.907i)7-s + (−0.440 + 2.96i)9-s + (1.03 + 0.337i)11-s + (2.26 − 4.44i)13-s + (−1.59 − 3.52i)15-s + (2.19 + 0.347i)17-s + (−2.39 − 3.30i)19-s + (−2.21 − 0.163i)21-s + (−2.13 − 4.18i)23-s + (4.19 + 2.71i)25-s + (4.39 − 2.77i)27-s + (0.981 + 0.712i)29-s + (0.992 − 0.720i)31-s + ⋯ |
L(s) = 1 | + (−0.653 − 0.757i)3-s + (0.959 + 0.283i)5-s + (0.343 − 0.343i)7-s + (−0.146 + 0.989i)9-s + (0.313 + 0.101i)11-s + (0.628 − 1.23i)13-s + (−0.411 − 0.911i)15-s + (0.532 + 0.0843i)17-s + (−0.550 − 0.757i)19-s + (−0.484 − 0.0357i)21-s + (−0.445 − 0.873i)23-s + (0.839 + 0.543i)25-s + (0.844 − 0.534i)27-s + (0.182 + 0.132i)29-s + (0.178 − 0.129i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16762 - 0.541287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16762 - 0.541287i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.13 + 1.31i)T \) |
| 5 | \( 1 + (-2.14 - 0.633i)T \) |
good | 7 | \( 1 + (-0.907 + 0.907i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.03 - 0.337i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.26 + 4.44i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 0.347i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (2.39 + 3.30i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.13 + 4.18i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.981 - 0.712i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.992 + 0.720i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 3.29i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (8.57 - 2.78i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.0645 - 0.407i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (13.7 - 2.18i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.76 - 11.6i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.344 + 1.05i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (1.22 - 7.74i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (6.04 - 8.32i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.93 + 3.02i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (5.56 - 7.66i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.79 + 17.6i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.10 + 12.6i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (15.3 - 2.42i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.54518466308028953090019341189, −10.68157946507399839511911660112, −10.05521105586436605776351796411, −8.617604121587083190979571806805, −7.66011101560686888163020297834, −6.52315413289014326013600807406, −5.85913996620450766621195957567, −4.70891162078864930309436754144, −2.76702278573533390354109192344, −1.23968184451913104005894868358,
1.71639216613743298259961495148, 3.71025141617754350365211730383, 4.89025669417410184257846358211, 5.86826168814586652051954307684, 6.59364443120937429837373448903, 8.326641696620081731578426102739, 9.295788971018529115407719491996, 9.895792918742870280317061436590, 10.95910708650898901189793257044, 11.75571523869275090713246423856